text:
Talk:Equivalence relation

Duplication of definition?
Should the three properties really be defined here or is it enough to refer to binary relation? Or should these properties have their own pages a la Eric Weisstein? -- Jan Hidders

 * I don't think the small amount of duplication is a problem, and it aids understanding.
 * PS: Hopefully, we can soon get up to the level of depth Weisstein had (unsigned comment by Gareth Owen 09:25, Jul 13, 2001 — Paul August ☎ 17:43, May 23, 2005 (UTC))

Ok. I would agree with that. But do you think that Wikipedia should define the same terms, i.e., if Eric has a definition of a certain mathematical concept then so should Wikipedia? Or should Wikipedia be more "course grained"? -- Jan Hidders

 * Duplication, in addition to aiding "understanding", also helps to make Wikipedia more robust, in that an error in one place can be corrected by looking somewhere else. Paul August ☎ 17:44, May 23, 2005 (UTC)

Content move
I moved this from the main page:

 * Given an arbitrary relation R, we define the equivalence relation ~ as follows:
 * (Reflexivity) Define x~x.
 * (Symmetry) Define x~y whenever R(x,y) *or* R(y,x) is true.
 * (Transitivity) Define x~y whenever there exists z such that R(x,z) and R(z,y) is true.

This is incorrect; the equivalence relation generated by R is a bit more complicated. In the third step, one has to allow for a whole sequences z1,...,zn of intermediate elements, and for each i one has to allow for R(zi,zi+1) *or* R(zi+1,zi). AxelBoldt 22:10 Nov 23, 2002 (UTC)

"in thermal equilibrium with"?
On what set is "in thermal equilibrium with" an equivalence relation? (unsigned comment by anon 69.202.74.207 — Paul August ☎ 17:43, May 23, 2005 (UTC))

Asymptotical equivalence
I did not (yet) find anything about the notion of asymptotical equivalence (x_n) ~ (y_n) iff  (x_n-y_n) = o(y_n) and the "crude" equivalence x=Theta(y) <=> x=O(y) and y=O(x). — MFH: Talk 17:14, 23 May 2005 (UTC)
 * PS: found the first one (slightly different and not completely equivalent definition) at Asymptotic... see Talk:Asymptotic analysis...  — MFH: Talk

Doesnt 2 and 3 imply 1?
Hmmmm. Let's see:

 * by 2, a~b, then b~a
 * take a=c
 * So, by 3, a~a

QED

Yes, I know this is wrong, but where's the error?

--User:Mdob

Given a fixed a, there need not exist a b such that a~b. For example, the empty relation is not an equivalence relation. It is symmetric and transitive but not reflexive. -- Fropuff 01:08, 12 October 2005 (UTC)

The definition of Equivalence Relation given in http://www.iscid.org/encyclopedia/Equivalence_Relation seems to say that the empty relation is an equivalence relation. It seems like the empty relation is vacuously reflexive to me. Is this wrong or are there just different interpretations?

 * I would think that an empty relation on an empty set would vacuuously be an equivalence relation, but that an empty relation on a non-empty set would not be an equivalence relation because it does not satisfy x ~ x for any x in that set. 129.110.241.254 21:18, 22 August 2006 (UTC)

Perhaps empty relation is not well defined. Define a relation ~ on the integers such that for all integers a and b, a~b is false. 2 and 3 are vacuously satisfied but 1~1 is false, so this is not an equivalence relation. The error in your proof is thus the assumption that there exists a, c such that a=c. If there is (well, if for all a there exists c such that a~c) then we have an equivalence relation. —Preceding unsigned comment added by 68.1.101.154 (talk) 07:23, 28 May 2008 (UTC)

This was an exercise in a logic class I had, find a relation that was symmetric and transitive but not reflexive. A simple example, as said before is the relation that maps all pairs to false. It is well defined. The reference book of the course was "Mathematical Logic", by Ebbinghau Flum and Thomas. The statement in the definition section that integers modulo 2 is not an equivalence relation is absolutely incorrect and I am going to change the section in the definitions to reflect this. Antares5245 (talk) 01:31, 7 June 2010 (UTC)

 * I misread the relation "a ~ b" iff a and b are both even as "a ~ b" iff a = b (mod 2), which isn't quite the same. Sorry about the misedit there.  JamesBWatson restored it correctly, although I'm still not sure that it should be in the definition section. Antares5245 (talk) 09:00, 12 June 2010 (UTC)

Partial Order : Lattice theory :: Equivalence relations : ?
I hereby invite someone to add a section on groups and equivalence relations. Lattice theory is the algebraic structure of partial orderings. What is the algebraic structure of equivalence relations? Bas Van Frassen (1989) convinced me that it is (non-Aberlian) group theory, but algebra texts are typically totally silent about this fact, with the possible exception of Birkhoff and MacLane (1979: 72). The mathematicians whom I've queried about this tell me they have no idea what I am talking about. A curious situation indeed... On equivalence relations and group theory, see and the references therein.202.36.179.65 10:22, 18 March 2006 (UTC)

I went to work and answered my own query.202.36.179.65 11:44, 8 August 2006 (UTC)

"Sibling of" is NOT Transitive
If it were, then it would lead to people being their own sibling. A sibling of B implies B sibling of A. Then by transitivity, A would be a sibling of A, which is false. It may sound like I am committing the error "2 and 3 imply 1" above, but I'm not. If the relation were transitive, then anyone that had a sibling would be considered a sibling of themself as well, which is clearly undesirable. —The preceding unsigned comment was added by 67.124.203.53 (talk • contribs) 17:07, May 25, 2006 (UTC)
 * Correct. Paul August ☎ 17:38, 25 May 2006 (UTC)

It depends on what you mean by "sibling." if we define "A is a sibling of B" to mean "A has the same two parents as B" then surely the sibling relation is an equivalence relation. -Misfit 20:18, 13 September 2007 (UTC)
 * This further illustrates the dangers in extending the idea of equivalence relation (a mathematical concept) to collections of objects and relationships between those objects which are not mathematical objects. I don't like the idea of the "set" of human beings because I'm not really sure that there is such a thing. Having examples which are not mathematical means that the concepts involved must be really well-defined, or they fall apart (as in the case of sibling). I have never seen the term "sibling" mean "having the same parents as;" in general, as far as I am aware, it means, "is a sister or brother to," and in this sense "sibling" would not be an equivalence relation, as has already been pointed out. Xantharius 03:37, 14 September 2007 (UTC)

The word "set" really ought to be able to include the set of human beings. I don't think this is the issue, and blocking things like that impedes understanding. The issue is that, as we have seen, sibling is not well-defined (or it is, but that definition is not generally agreed upon, which is perhaps worse). We shouldn't block real-world examples, just be more careful with them. —Preceding unsigned comment added by 68.1.101.154 (talk) 07:25, 28 May 2008 (UTC)

That's been sitting there for years? How ridiculous! I've changed it, because it was completely wrong.24.58.63.18 (talk) 22:12, 15 March 2009 (UTC)

Just a thought but sibling is transitive and here's why:

If relation R is Transitive "xRy and yRz implies xRz"

When y = x we have "xRx and xRz implies xRz", "xRx" is fully determined by the reflexive property of relation R. If R is irreflexive x is never related to x by R, hence R being transitive tells use nothing about "xRz". If R is reflexive "xRx" is always true and we have "xRz implies xRz" hardly illuminating for obvious reasons.

The knowledge that relation R is transitive adds information when x != y and nothing when x = y.

When y = z we have "xRz and zRz implies xRz", again "zRz" is determined by the reflexive property of relation R. If R is irreflexive z is never related to z by R, hence R being transitive tells use nothing about "xRz". If R is reflexive "zRz" is always true and we have "xRz implies xRz" again hardly illuminating for the same obvious reasons.

Hence knowing that relation R is transitive adds information when y != z and x != y and nothing when x = y or y = z.

When x != z we find that knowing that relation R is transitive "xRy and yRz implies xRz" is most illuminating.

Now when x = z we have "xRy and yRx implies xRx", If R is reflexive "xRx" is always true and x = z is ok.

However, iff R is irreflexive "xRx" can never be true, hence x = z is false and x != z.

It seems to indicate that when R is irreflexive and transitive, x != z and the transitive property is meaningless when x = z. In other words when x = z "xRy and yRx implies nothing more than the two individual statements themselves imply" and it is NOT CORRECT to say that relation R is NOT Transitive. Note further that the above applies when relation R is symetric, anti-symetric or otherwise.

Now let's take a more concrete example:

Let x = John, y = Mary, z = Susan and R be the relation sibling.

John and Mary are siblings.

"John sibling Mary" and "Mary sibling John" implies "John and Mary are siblings" and nothing more.

Clearly when "John sibling Mary" is known then "Mary sibling John" adds nothing to the discourse because "Mary sibling John" is the same fact not two different facts.

now suppose the following is also true

Mary and Susan are siblings

then "John sibling Mary" and "Mary sibling Susan" implies "John sibling Susan" which adds to the discourse and is the meaning of the transitive property in the first place. [edit] —Preceding unsigned comment added by 64.231.95.9 (talk) 19:59, 11 November 2009 (UTC)

This is not transitive.

If John sibling Mary, then Mary sibling John. By transitivity, John sibling John. Which we already defined to be not part of the relation. Therefore it is NOT transitive. This wikipedia article is the reason why half my math class got this wrong. — Preceding unsigned comment added by 209.6.94.214 (talk) 07:25, 12 February 2012 (UTC)

 * Why is this article to blame for half your class getting it wrong? It states quite clearly that “sibling” is not an equivalence relation, and it explains why. So the reason must lie elsewhere, is my guess. Hanche (talk) 10:46, 12 February 2012 (UTC)

The article states that is a sibling of is transitive on set of 3 distinct people A,B,C. Since this is not the same as transitive I find the example misleading, and I think it should be removed.80.216.62.225 (talk) 08:53, 15 August 2012 (UTC)

transtive, symmetric and irreflexive
The following doesn't make any sense:--345Kai 23:30, 7 September 2006 (UTC)

 * Relations that are transitive, symmetric, but irreflexive are very rare (e.g., the empty relation). The previous example illustrates why. More generally, if a relation R is symmetric and transitive, then for any x, y in the field of R, if xRy, then yRx (by symmetry), and xRx (by transitivity). Hence if there exist an x and y in the field of R, such that y≠x and xRy both come out true, the reflexivity of R follows and R must be an equivalence relation. R can be transitive, symmetric but irreflexive iff x is an "island," namely for an x that is not related to anything at all.

What you are objecting to is legacy material I found when I began my major edit of this entry. Feel free to edit or delete.202.36.179.65 14:59, 5 November 2006 (UTC)

From order to equivalence via symmetry
I think this paragraph is incomprehensible, misleading and even wrong I think it's better to remove it... — MFH:Talk 21:41, 16 October 2006 (UTC)
 * what could be a "reflexive strict order" ?? strict means non-reflexive !
 * what means "antisymmetry enhanced to full symmetry" ??

 * Here's what I meant. Begin with a strict order. Then replace non-reflexivity with reflexivity and the result is an equivalence relation. Likewise, start with a partial order, then replace antisymmetry with symmetry to obtain, again, an equivalence relation. It is also valid to go the other way. A variety of of types of relations can be derived from an equivalence relation by suitably relaxing parts of the definition of an equivalence relation. I see relation types as forming a hierarchy in a manner similar to the way normal modal logics form a hierarchy. Equivalence relations are at the top of the relations hierarchy is a manner analogous to the way S5 is at the top of the normal modal logic hierarchy.202.36.179.65 15:11, 5 November 2006 (UTC)

 * you say "Begin with a strict order. Then replace non-reflexivity with reflexivity", but is that well-defined? In a 2-element set {a, b} if I define the strict partial order to be (a<b) (and thus also ~(b<a)) how do I make that reflexive? "~(b<a) and ~(a<b)" or "(b<a) and (a<b)"? Of course I end up with two equivalence relations, but they are different. If for a general set with s.p. order I have to find its ``corresponding`` equivalence relations, then what algorithm do I follow? Is the answer different from the set of all possible equivalence relations on that set? --MarSch 12:28, 27 November 2006 (UTC)

 * I rewrote parts of section 2 wishing to satisfy your objections. Does that section now meet with your approval? In any event, above I should have written "Begin with the concept of a strict order" and so on. You ask " If for a general set with s.p. order I have to find its ``corresponding`` equivalence relation, then what algorithm do I follow?" Given a concrete relation defined over a specific set, the answer is that there is no such algorithm.202.36.179.65 09:37, 7 March 2007 (UTC)

 * huh? What is being talked about here?  Certainly you can union the diagonal (x,x) with whatever relation you begin with.  Now it is reflexive.  Similarly you can union with the reflection through the diagonal, thus you can make symmetric.  You can take transitive closure, thus making the smallest transitive relation containing the one you begin with.  Depending on starting properties, and the order you add pairs to your relation, you may end up with an ER.  For example, for a SPO it is not enough to "reflexivise" then "symmetrize."  You need to "transitivise" again!  Maybe you are talking about something else, I don't know.24.58.63.18 (talk) 22:25, 15 March 2009 (UTC)

Some examples may indeed be confused and confusing
Most of the Examples are material I found when I began my major rewrite of this entry. More than one Example I cannot follow myself. My main contribution to the examples is pointing out that the usual number systems can all be defined as equivalence relations. My only contributions to the examples of non-equivalences are attempts to clarify by rewording, and those attempts are not a reason for any of you to hold your fire! In any event, those wishing to heavily edit or delete an example should feel free to do so.202.36.179.65 15:28, 5 November 2006 (UTC)
 * Please at least one sane example to this article... like taking set of {1,2,3} and building equivalence relations {(x0,y0),(x1,y1)...} using this set. Consider that most readers of this article will be math students like myself. Finding pictographs with obscure meaning is absolutely unhelpful. — Preceding unsigned comment added by 79.183.12.73 (talk) 17:15, 1 May 2012 (UTC)

More concrete example of transitive, symmetric, but irreflexive
I think that the example (quote begins) (quote ends) could be supplied by the concrete case of the relation a R b = "a and b are both even numbers" on the set X of natural numbers (all natural numbers, not just the even numbers). This relation is clearly symmetric (if a and b are both even, b and a are both even), it is transitive (if a and b are both even and if in addition b and c are both even, a and c are both even), but it is not reflexive because 3 and 3 are not both even numbers. 129.132.146.184 09:13, 27 February 2007 (UTC)
 * The empty relation R on a non-empty set X (i.e. aRb is never true) is vacuously symmetric and transitive, but not reflexive (except when X is also empty).
 * Yup, that works. Paul August ☎ 02:37, 8 March 2007 (UTC)
 * not really. Not considering the headline is "irreflexive." The only relation with those 3 qualities is the empty relation.24.58.63.18 (talk) 22:30, 15 March 2009 (UTC)
 * Your example is a good one for 'transitive, symmetric, but not reflexive'. Unfortunately, 'irreflexive' doesn't mean 'not reflexive', it means 'never reflexive'. Adammanifold (talk) 21:27, 14 May 2009 (UTC)

A concrete example of an irreflexive, symmetric and transitive relation is Sibling(x,y) where x and y are elements of the set Person.

A commonly accepted definition of Sibling(x,y) is that x and y have the same parents and that x and y are distinct individuals.

The Sibling relation is clearly irreflexive; for all x an element of Person, Sibling(x,x) is false, by definition x is not a sibling of him or herself.

It is clear that the Sibling relation is symmetric; Whenever x and y are siblings, y and x are siblings.

It is also clear that the Sibling relation is transitive; Whenever (x and y are siblings) and (y and z are siblings), it is also true that (x and z are siblings).

Whenever a relation has the distinct individuals clause in its definition, it is irreflexive. Hence, it should be easy to find many examples with the above stated properties. —Preceding unsigned comment added by 64.231.125.50 (talk) 14:09, 10 November 2009 (UTC)

 * The problem is that such a relation isn't transitive either. 2A01:CB10:21:8500:F110:46D6:DB03:EAAB (talk) 21:56, 23 July 2022 (UTC)
 * That is right. Also, 24.58.63.18's post of 15 Mar 2009 above is right: if R is transitive, symmetric, and non-empty, then aRb holds for some a,b. By symmetry, bRa also holds, Hence, aRa by transitivity from aRb and bRa. That is, R cannot be irreflexive. - Jochen Burghardt (talk) 14:44, 24 July 2022 (UTC)

Equivalence relations on objects which are not sets
The article, as way of introduction to the idea of equivalence relation, cites examples of equivalence relations on the "set" of all human beings, and on physical objects. This has been raised previously, but nothing was done. I agree that in order to explain the idea of equivalence relation that these could be cited in a section of analogies of equivalence relations. However, since as far as I am aware neither human beings nor physical objects are members of any set that I am aware of (possibly they are, however, a class) then this should be removed or edited. Unless there are any objections, I shall do exactly that. Xantharius 20:32, 25 June 2007 (UTC)

Please, have a look at Set and in particular the words "any collection of distinct objects considered as a whole", and tell me how they do not apply to human beings or other physical objects. Thus, despite my lack of advanced mathematical compentences, I object (belatedly, also thereby further declaring that I belong to the set of people who object, and in fact the set of objects that object!) and kindly suggest your removal be undone. --Lasse Hillerøe Petersen 20:48, 16 July 2007 (UTC)

 * Reading further on in the article for set, we see that it has to be possible, for a collection to be a set, for us to be able to determine whether something is in or out of the set. Well, the idea of set might apply to human beings (maybe), but definitely does not apply to all physical objects, because not all physical objects can be well-defined, necessary for the idea of set to apply. There is no such thing as the set of all clouds, for example, because sometimes it's not possible to tell what is a cloud, and what isn't (How many clouds are there? Is that a cloud, or just some smoke? Are those clouds touching, or not? Does that make them two clouds, or just one?), and I think a cloud counts as a physical (if somehwhat nebulous) object. You might even want to say "solid objects", but then what counts as solid? While it might be argued that a cube of steel is solid, it's not solid if you heat it up high enough. Is the earth a solid object? What about a gel? It's very hard to be precise.

 * On the subject of humans, I wonder if an unborn baby is considered a human (I would hope so) and thus the equivalence relation "has the same birthday as" on the set of all humans is not well-defined, since there are some humans without birthdays.

 * Although the colours of the French flag are (possibly) well-defined (see discussion on set), defining objects to be collected is extremely difficult if there is any imprecision as to what exactly counts as defining the object. What colour is red, exactly? We all instinctively know what red is, and even what red is used in the flag, but can I talk about the set of all colours used on flags? I don't think so, because there is disagreement as to exactly what kinds of red are used on different flags.

 * This all might seem very picky, but I think I've already shown that equivalence relation does not apply to the "set" of all physical objects or the "set" of all humans. Yes, thermal equilibrium and birthdays are analogues of these ideas for non-mathematical scenarios, but they not equivalence relations proper. I have taken the idea of set as applying to collections of mathematical objects. With some work, I suppose that other ideas could be used. I'm going to leave it as it is, but feel free to have a go at precisely defining these ideas for humans and physical objects if you can. Xantharius 23:47, 16 July 2007 (UTC)

It would seem you are saying that because there can be ambiguous or unclear definitions of "human beings" or various other physical objects, it is impossible to talk about sets of any such objects. I think that conclusion is extreme. And I don't understand why you want to impose a different and narrower "standard" for what can constitute a set here, when it obviously doesn't agree with the definition on the Set page. At least be persistent and try have that page changed too: certainly the definition of what can constitute a set and what can not is no less important there than here, right?

(BTW, "same birthday as" can be well-defined if for example the unknown birthday is included as a possible birthday. And that's just one way to do it.)--Lasse Hillerøe Petersen 12:18, 17 July 2007 (UTC)

 * I understand your points. I think I would first state that I was coming from the point of view of axiomatic (rigorous) set theory, wherein there is no such thing as the set of humans because humans or collections of them cannot be constructed from mathematical ideas.

 * My main objection, on this basis, is the that the formal definition of "set" (see axiomatic set theory) is, in mathematics, actually a very narrow concept. Although the article on set says that one definition of set is "any collection M of definite, distinct objects m of our perception or of our thought (which will be called the elements of M) into a whole", this is a definition from naive set theory. In axiomatic set theory there is no such thing (and this is not my opinion by the way: it just doesn't exist) as the "set of all sets", or "the set of all groups". These things are too large to be sets: they are instead classes. And, indeed, if one knows about how sets really are constructed, it becomes apparent quite quickly that sets of things that aren't mathematical objects to begin with is impossible.

 * If we are going to stick with naive set theory, though, I suppose that there might be such a thing as the "set of all people". As far as physical objects go, however, that's a definite no, as I think it impossible to say for sure what constitutes and what does not constitute a physical object. I could consider that I am a physical object. And my hot steak on a plate might be, too. We are not in thermal equilibrium with each other since we are at different temperatures. If I eat the steak, are these objects now in thermal equilibrium, or is there now only one object? What happened to the steak? Is it in thermal equilibrium with me, or not? If one can't say yes or no, then the objects are not members of any real set which is going to be of use

 * I do take your point on the "set of all people", though, and will put this back in some form, as long as it understood that this comes from naive set theory. Xantharius 15:26, 17 July 2007 (UTC)

 * I think you are inviting a lot of philosophical rhetoric that is not needed in this context. How many physics and math articles would need to be rewritten to accommodate the the philosophical objection that objects in the real world are not well-defined?  And why stop there?  We could preface every article on WP with a few paragraphs on the problems of natural language and its referents.  One reason it isn't even hypothetically possible to do this is that without some naive set theory you cannot do mathematics, (and without some naive understanding of our world you cannot build natural language), since you're not even sure what it means to be a symbol, let alone distinct symbols.  When people first start learning about axiomatic set theory they make this mistake of assuming the mathematics can do itself, without any interpretation, all completely formally and automatically, but it cannot of course.  I think it is implicit in "the set of human beings" that in fact what is being discussed is a model where it is unambiguous to determine whether something is a human or not a human.  Most of us are comfortable thinking in this model, indeed it is necessary in order to communicate.  —Preceding unsigned comment added by 24.58.63.18 (talk) 23:24, 15 March 2009 (UTC)

Rational numbers
Just a quick note: The relation (a,b) equivalent to (c,d) if ad=bc on N x N does not give all rational numbers, but only the positive rational numbers. To get all rationals you have to look at this relation as defined on Z x Z^*, but I'll leave it up to others to make this precise in the article. —Preceding unsigned comment added by 136.152.181.253 (talk) 17:40, 19 October 2007 (UTC)

conjugacy
What about complex, topological or conformal conjugacy ? Is it a Equivalence relation ? --Adam majewski (talk) 08:11, 31 January 2009 (UTC)

Topological conjugacy is an Equivalence relation ( see Scholarpedia )

subs --> subsets?
if this isn't a shorthand, please wikilink it. Thanks! —Preceding unsigned comment added by 128.32.35.76 (talk) 00:46, 13 August 2009 (UTC)
 * Just random vandalism. Algebraist 00:51, 13 August 2009 (UTC)

Euclid anticipated equivalence
This heading is clearly a complete fabrication and should be deleted. Even the editor who added it admitted to giving Euclid a very generous interpretation. Unless I hear strong objections to the deletion of the section in question I will delete it promptly. ReneGMata (talk) 14:46, 13 January 2010 (UTC)

 * I agree that in its present form the section is a stretch. However, "complete fabrication" seems a little strong, and it may be worth a mention that Euclid did make use of (essentially) the idea of equivalence in his deductive system. Perhaps a drastic rewrite and abbreviation, rather than total deletion, but I don't have time to do it now. JamesBWatson (talk) 17:03, 13 January 2010 (UTC)

It would be very nice if this section could be supported by reasonable citations or references. In fact it would be very nice if all the editors of Wikipedia adhered to facts rather than speculation. And yes, I believe that "stretching" (your word) is a form of speculation. And no, I do not think that a re-write of speculation will convert it into fact. ReneGMata (talk) 18:08, 14 January 2010 (UTC)

 * I was not suggesting rewriting speculation: I was suggesting rewriting the element of truth in there. Euclid certainly did use axioms which are now part of the definition of an equivalence relation, and I see no reason not to mention the fact. JamesBWatson (talk) 14:57, 18 January 2010 (UTC)

History
Euclid aside, where is the first recorded explicit defnition of equivalence relation? I think it would be worth adding to the article. Simplifix (talk) 12:15, 10 February 2010 (UTC)
 * The relation of equipollence used by Giusto Bellavitis is an early example with a suggestive label.Rgdboer (talk) 23:01, 3 November 2014 (UTC)

Relations seems to be defined for other classes than "sets"
Jacobson mention that monoid isomorphisms define an equivalence relation (in Basic Algebra I, p. 37). So, is he considering "the set of all monoids"? Is this really a set? (is Mon a small category?)

If not, there should be a note about equivalence relation over more general classes? Helder (talk) 21:32, 13 May 2010 (UTC)

I too have been feeling like relations should be defined on classes. Doesn't the usual definition of cardinal numbers rely on partitioning the (proper) class of all sets into equivalence classes? That might be important enough to include as a blurb. Rschwieb (talk) 00:47, 26 July 2011 (UTC)

Theorem on projections partially wrong
It is currently written:
 * Theorem on projections: Let the function f: X → B be such that a ~ b → f(a) = f(b). Then there is a unique function g : X/~ → B, such that f = gπ. If f is a surjection and a ~ b ↔ f(a) = f(b), then g is a bijection.

I think the last sentence is wrong, and here is a counter-example. Let the set X have two elements. Let the equivalence relation be the equality, i.e. a ~ b ↔ a = b. Let the set B be a singleton. The function f: X → B is a constant function and is a surjection. But g : X/~ → B can not be a bijection since X/~ has two elements but B only one. Frozsyn (talk) 12:45, 16 November 2010 (UTC)

 * In your example, if f is a constant function from a two element set {a,b} to a one element set, since a is not equivalent to b, f does not have the property that a ~ b ↔ f(a) = f(b). That extra property in the last sentence is important. — Carl (CBM · talk) 12:55, 16 November 2010 (UTC)

 * Thanks for the reply! Somehow, I re-read the statement many times without noticing the property a ~ b ↔ f(a) = f(b). I assumed only a ~ b → f(a) = f(b) as written in the first part. My bad. Frozsyn (talk) 14:16, 16 November 2010 (UTC)

Isn't it ≈ & NOT ~
Hmmmmmmm? Alt 247 —Preceding unsigned comment added by 173.218.85.222 (talk) 20:06, 16 February 2011 (UTC)
 * No. Algebraist 20:07, 16 February 2011 (UTC)

Imprecise definition of partition
A partition of X is a set P of nonempty subsets of X such that every element of X is an element of a single element of P.
 * Correct, fixed. McKay (talk) 01:47, 21 February 2011 (UTC)

Projection (relational algebra)
The link to the Projection (relational algebra) (main article) in the "Projection" chapter, is it relevant at all? Should not the "projection" be called the "quotient map"? I do not see how projections and quotient maps are related. --Beroal (talk) 22:59, 1 March 2011 (UTC)

Poor lede
The current lede is unmathematical and suggests segregation:
 * In mathematics, an equivalence relation is the relation that holds between two elements if and only if they are members of the same cell within a set that has been partitioned into cells such that every element of the set is a member of one and only one cell of the partition. The intersection of any two different cells is empty; the union of all the cells equals the original set. These cells are formally called equivalence classes.

The fact that equivalence classes are the outcome of an equivalence relation is reported deep in the article. As it stands, the article encourages the enemies of social equality who would perpetuate social classes such as the caste system in India. There are elements that feed off class conflict and the lede of this article, which should be on mathematics, suggests that division is natural. The lede should be rewritten as an article on binary relations, with the partition reserved for a conclusion following from the three intersecting properties.Rgdboer (talk) 22:31, 20 July 2015 (UTC)
 * Are you serious? — Arthur Rubin  (talk) 20:15, 2 January 2016 (UTC)

The impression was strong enough that it led me to make edits at Social equality in August. — Rgdboer (talk) 21:08, 2 January 2016 (UTC)

The following is my suggestion for the lede:
 * An equivalence relation is a binary relation that is at the same time a reflexive relation, a symmetric relation and a transitive relation. As a consequence of these properties an equivalence relation provides a partition of a set into equivalence classes.

As written, the current lede (still as quoted above) describes the Partition of a set, not a binary relation upon it. Since 93 Watch this article, discussing a proposed change of lede before posting may produce a consensus. — Rgdboer (talk) 00:53, 3 January 2016 (UTC)
 * It's an improvement; I think, but, per WP:TECHNICAL, it might be a good idea to include the definition of partition of a set in the lead, as as well. I think the following is a little better, but I'm a professional mathematician, so I'm familiar with the terms involved.  How about:
 * "An equivalence relation R over a set X is a binary relation over X that is reflexive, symmetric, and transitive. As a consequence of these properties, an equivalence relation R corresponds to a partition of X."
 * I think that's both accurate and understandable, but I'll defer to others on "understandable". — Arthur Rubin  (talk) 06:12, 3 January 2016 (UTC)
 * We still need to introduce equivalence classes in the lede, though. I'm just proposing a first sentence. — Arthur Rubin  (talk) 06:19, 3 January 2016 (UTC)

In using WP:Pipes it says "keep links as simple as possible." For instance, Partition is a disambiguation page listing 7 mathematical uses of the term so your pipe of Partition of a set leads a reader to underestimate the expanse of the term. Similarly piping the three relation links to their adjectives makes the reading smooth, but detracts from the emphasis on Relation. These comments pertain to internet linguistics and hyperlink grammar, recent developments with differing opinions. However, since the two versions are nearly identical except for these points, the change is due. — Rgdboer (talk) 21:31, 7 January 2016 (UTC)
 * I can see your point about partition of set, although it makes the sentence slightly incorrect. I disagree about use of relation, though.  My version is exactly the way it would be written in a math textbook, is understandable, and has the proper links.  The only people hurt are offline readers who don't understand the basic material, and it doesn't hurt them much.  — Arthur Rubin  (talk) 06:55, 9 January 2016 (UTC)

What do you mean, "...it makes the sentence slightly incorrect" ? — Rgdboer (talk) 20:29, 9 January 2016 (UTC)

Cell
The technical term 'cell' appears to have no entry in the disambiguation page for 'cell'. Jim Bowery (talk) 22:17, 1 January 2016 (UTC)
 * I've never seen "cell" used in this context in the mathematical literature. A quick Google search finds only a few Wikipedia articles and some quotes from Wikipedia.  Objects in "cell"s or "bin"s are sometimes used as examples of equivalence relations.  — Arthur Rubin  (talk) 20:13, 2 January 2016 (UTC)
 * The Encyclopedia of Mathematics has no such entry. I think it may be original to Wikipedia.   — Arthur Rubin  (talk) 20:25, 2 January 2016 (UTC)

 * "Cell" and "equivalence relation" are an unusual combination, but "cell" and "partition" (of a set) are often used together. If you search at Scholar for "cell of the partition" and "cells of the partition" (with quotes) you'll find many examples. Given that partitions and equivalence relations are more or less the same concept, that's enough to justify the mention of cells here, imo. McKay (talk) 01:24, 17 March 2016 (UTC)

Math style
@: WP:MOSMATH says that formatting should not be changed without clear improvement or consensus. Your edit introduced a mix of formattings, as you changed only a part of the article; this was certainly no improvement. For this reason, I reverted your edit. Apart from that, there are problems with the format, such as ugly rendering in Png mode, and MathJax mode sometimes having its caveats, if I remember right. As far as I know, using

formatting is the most promising alternative to both Html and. - Jochen Burghardt (talk) 22:02, 15 March 2016 (UTC)

 * OK. And thanks.
 * I did not know that there are problems with <math> tag and mathjax.Thought that this is the best way to display math currently.
 * Rushikeshjogdand1 (talk) 04:31, 16 March 2016 (UTC)

External links modified
Hello fellow Wikipedians,

I have just modified 1 one external link on Equivalence relation. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:
 * Added archive https://web.archive.org/web/20130509233055/http://planetmath.org/equivalencerelation to http://planetmath.org/equivalencerelation

When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.

Cheers.— InternetArchiveBot  (Report bug) 07:10, 25 December 2016 (UTC)

External links modified
Hello fellow Wikipedians,

I have just modified one external link on Equivalence relation. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:
 * Added archive http://arquivo.pt/wayback/20160514115224/http://www.bangor.ac.uk/r.brown/topgpds.html to http://www.bangor.ac.uk/r.brown/topgpds.html

When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.

Cheers.— InternetArchiveBot  (Report bug) 05:56, 22 September 2017 (UTC)

Major Mistake
Skimming this article at 9:01 am, US Central Time, on Friday, July 23rd, 2021, I found the following mistake. I'd correct it immediately if I knew how to do that, but regrettably, I don't know how to edit or write formulas on Wikipedia. In any case, here is the error:

Symmetric and transitive: The relation R on N, defined as aRb ↔ ab ≠ 0. Or any partial equivalence relation;

Transitive means that if aRb and bRc, then aRc. Consider a=2, b=3, and c=2. This counter-example proves that ≠ is not transitive, because 2≠3 is true, and 3≠2 is true, yet 2≠2 is false.

We therefore require a replacement relation that is symmetric and transitive but not reflexive. I would recommend a binary relation that I call the "department head" relation. What ever request you make, the answer is always "no." In other words, for all x and for all y, xRy is false. This binary relation is not reflexive, but it is symmetric, and it is transitive. (Any one who has ever worked in education will find this example intuitive and familiar.)

Respectfully submitted, Gregorybard (talk) 14:09, 23 July 2021 (UTC)

 * You are right that ≠ itself isn't transitive. However, R defined by aRb ↔ ab ≠ 0 is transitive. (Note that R is different from ≠, e.g. 2R2, but not 2≠2, and e.g. 0≠1, but not 0R1.) The definition of R can be re-phrased as aRb ↔ a≠0 ∧ b≠0. Then aRb and bRc implies a≠0 and b≠0 and c≠0 by the alternate definition, hence a≠0 ∧ c≠0, hence aRc. BTW: your "department head" relation is known as the empty relation. I feel it is too particular to be the only example for a partial equivalence. - Jochen Burghardt (talk) 14:42, 23 July 2021 (UTC)
 * I find this example and the one given in the article interesting. However they both look like ternary relations in disguise (between a, b and 0, in this case). Which seems inevitable for a non-empty non-reflexive relation to be both symmetric and transitive. I don't know if this is true, have no idea how it could be proved and realize that it doesn't make the example any less correct or relevant. I just thought it would be cool if someone more knowledgeable could explain why. 2A01:CB10:21:8500:CD8:3FB4:5BD5:E3AE (talk) 11:28, 25 July 2022 (UTC)
 * I'd like to spend some effort on trying to prove your conjecture - if I'd understand what exactly it consists of. Maybe "Every non-empty non-reflexive relation that is both symmetric and transitive is a ternary relation in disguise." But then, how to formalize "ternary relation in disguise"? - Jochen Burghardt (talk) 17:52, 25 July 2022 (UTC)

Transitive Relation
If (a, b), (b, c) (a, c) belongs R, and (b, b) doesn't belongs to R then will it be transitive or not. If not then why? 2409:4063:420B:CD60:75BF:7362:1CFF:DB7E (talk) 05:12, 5 December 2021 (UTC)

Composition notation
Writing the composed relations R and S in the backward fashion of function composition is not recommended. A section using calculus of relations institutes this notion. It ignors the left to right reading of relations. Seeking consensus here to repress this perverse initiative. The greater than (>) and less than (<) relations show importance of direction of text. Habits from calculus of real and complex variables need reformation when addressing relations. Rgdboer (talk) 04:05, 8 June 2022 (UTC)

"simple" example - what are the square brackets?

 * What are the square brackets in the "simple example" and why are these equivalent classes?
 * Quote (more or less):
 * On set X={a,b,c}, R={(a,a),(b,b),(c,c),(b,c),(c,b)} is an equivalence relation.

 * The following sets are equivalence classes of this relation:

 * [a]={a}, [b]=[c]={b,c}. [a]={a}, [b]=[c]={b,c}.
 * פשוט pashute ♫ (talk) 18:55, 16 November 2022 (UTC)

Evidently [–] assigns an element of the set to its equivalence class. Rgdboer (talk) 01:23, 17 November 2022 (UTC)

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Talk:Equivalence relation


text:
Talk:Eindhoven

Untitled
Still to be incorporated:

Schools: Eindhoven University of Technology, Fontys, Design Academy, Lorentz Casimir Lyceum, Eckart college, Augustinianum, Christiaan Huygens College, ISSE= International Secondary School Eindhoven

Sports club especially football, but also other sports such as swimming (Olympic champions such as Inge de Bruin and Pieter van de Hoogenband?): PSV.

Museums: Van Abbe museum, Prehistoric village, DAF museum.

Newspaper: Eindhovens Dagblad.

River: Dommel.

The Evoluon used to be a museum of technology. It currently serves as a congress centre. See http://www.evoluon.nl for details.

Sports: Tongelreep, numerous sport facilities in Woensel and Tongelre.

Shopping: the Heuvelgalerie, weekly markets.

Theatre: Zuidelijk Toneel, Philips Music Centre, Theater Het klein.

Going out: the Effenaar, the Danssalon, Pop-Ei.

Architectural highlights: Philips Stadion, Evoluon, numerous buildings have been declared Industrial Heritage, Witte dame, Bruine heer, Catharinakerk. — Preceding unsigned comment added by 212.153.190.xxx (talk) 11:09, 13 July 2001 (UTC)

The current building-up of the High Tech Campus Eindhoven, on the site of the Philips Research center (NatLab). It is owned still by Philips, but the philosophy of "open innovation" means that other companies are being invited to come and share the Philips labs facilities (cleanroom etc). (link to HTCE website). Should be coherently included at some point because this is an important technological focus in The Netherlands. — Preceding unsigned comment added by Insibato (talk) 09:43, 13 September 2005 (UTC)

Things to add
I do not live far from Eindhoven. So if you want to know anything about Eindhoven, write it to me. Talk page User Dgb

"PSV Eindhoven is the major football (soccer) club in the city. Playing in the Philips Stadion it is the dominant club in the Dutch highest league after AFC Ajax Amsterdam, named Eredivisie."

According to results, PSV is much better then Ajax the last 5 years. Change it in: "PSV Eindhoven is the major football (soccer) club in the city. Playing in the Philips Stadion it is a dominant club in the Dutch highest league, named eredivisie." I guess someone from Amsterdam wrote it.

Are the metro and urban figures based on anything at all? CBS does not seem to record them. —Preceding unsigned comment added by 94.212.44.210 (talk) 17:36, 18 May 2009 (UTC)
 * The urban figure is based on the former regional cooperation authority Agglomeratie Eindhoven, which includes all neighboring municipalities, Valkenswaard and Helmond. The metro figure refers to the present day regional authority SRE. That figure is a bit exaggerated, because most of the area has a functional connection to Eindhoven, but does not really have a metropolitan feeling... Pepijnk (talk) 09:19, 26 May 2009 (UTC)

IPA name
We need also an IPA representation of the name. —The preceding unsigned comment was added by 87.8.6.23 (talk) 18:30, 5 May 2007 (UTC).

Google Maps plugin?
I've added some Google Maps static map links, which display maps with some notable landmarks in Eindhoven. But this is (and I'll be as polite as I can without losing accuracy here) a piss-poor solution. Does anybody know if and when the Google Maps plugin for MediaWiki will be available on Wikipedia? -- BenTels (talk) 09:58, 30 August 2010 (UTC)

Tunas
I've added some information on the Tuna Ciudad de Luz and La Tuniña. But I seem to recall there was a third one as well, but I cannot remember which. Can anybody assist? -- BenTels (talk) 14:29, 30 August 2010 (UTC)

Statistics edited by User:81.205.211.100
This user has changed a lot of statistics and even removed a citation in favour of another number. Should these be reverted? Some of them are definitely wrong like that Eindhoven has over six hundred thousand people. That's not true at all. CodeCat (talk) 10:31, 9 October 2011 (UTC)

Timeline of Eindhoven
What is missing from the recently created city timeline article? Please add relevant content! Contributions welcome. Thank you. -- M2545 (talk) 11:17, 18 October 2015 (UTC)

External links modified
Hello fellow Wikipedians,

I have just modified 7 one external links on Eindhoven. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:
 * Added archive https://web.archive.org/web/20110722103029/http://www.intelligentcommunity.org/index.php?src=news&srctype=detail&category=Partner%20News&refno=582 to http://www.intelligentcommunity.org/index.php?src=news&srctype=detail&category=Partner%20News&refno=582
 * Added archive https://web.archive.org/web/20110726181907/http://www.intelligentcommunity.org/index.php?src=news&srctype=detail&category=Partner%20News&refno=629 to http://www.intelligentcommunity.org/index.php?src=news&srctype=detail&category=Partner%20News&refno=629
 * Added archive https://web.archive.org/web/20100328175502/http://eit.europa.eu/home.html to http://eit.europa.eu/home.html
 * Added archive https://web.archive.org/web/20110724160416/http://www.kunstlichtkunst.nl/en/Press/Press_information?action=viewnews&id=1321&year=2010 to http://www.kunstlichtkunst.nl/en/Press/Press_information?action=viewnews&id=1321&year=2010
 * Added archive https://web.archive.org/web/20110724171203/http://w3.tunacl.nl/index.php?id=en to http://w3.tunacl.nl/index.php?id=en
 * Added archive https://web.archive.org/web/20110724171204/http://www.stud.tue.nl/~tunina/index%20Engels.html to http://www.stud.tue.nl/~tunina/index%20Engels.html
 * Added archive https://web.archive.org/web/20140714151106/http://www.cncd.fr/frontoffice/bdd-region.asp?action=getRegion&id=4 to http://www.cncd.fr/frontoffice/bdd-region.asp?action=getRegion&id=4#tabs3

When you have finished reviewing my changes, please set the checked parameter below to true or failed to let others know (documentation at ).

Cheers.— InternetArchiveBot  (Report bug) 12:50, 21 December 2016 (UTC)

External links modified
Hello fellow Wikipedians,

I have just modified 10 external links on Eindhoven. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:
 * Added archive http://webarchive.nationalarchives.gov.uk/20070706011932/http://www.raf.mod.uk/bombercommand/dec42.html to http://www.raf.mod.uk/bombercommand/dec42.html
 * Added archive https://web.archive.org/web/20101022225733/http://www.rvk.dse.nl/ to http://rvk.dse.nl/
 * Added archive https://web.archive.org/web/20110613204248/http://www.ad.nl/ad/nl/1012/Binnenland/article/detail/2443794/2011/06/10/Rotterdam-onveiligste-gemeente.dhtml to http://www.ad.nl/ad/nl/1012/Binnenland/article/detail/2443794/2011/06/10/Rotterdam-onveiligste-gemeente.dhtml
 * Added archive https://web.archive.org/web/20100408230745/http://www.sre.nl/web/show to http://www.sre.nl/web/show
 * Added archive https://web.archive.org/web/20140313190025/http://www.elat.org/ to http://www.elat.org/
 * Added archive https://web.archive.org/web/20100306174018/http://www.ed-adverteren.nl/verspreidingsgebied/commercieel/edd to http://www.ed-adverteren.nl/verspreidingsgebied/commercieel/edd
 * Added archive https://web.archive.org/web/20150402143354/http://www.ns.nl/vertrekstaten/ehv4.pdf to http://www.ns.nl/vertrekstaten/ehv4.pdf
 * Added archive https://web.archive.org/web/20150402115817/http://www.ns.nl/vertrekstaten/ehv2.pdf to http://www.ns.nl/vertrekstaten/ehv2.pdf
 * Added archive https://web.archive.org/web/20150402132322/http://www.ns.nl/vertrekstaten/ehv3.pdf to http://www.ns.nl/vertrekstaten/ehv3.pdf
 * Corrected formatting/usage for http://www.cncd.fr/frontoffice/bdd-region.asp?action=getRegion&id=4

When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.

Cheers.— InternetArchiveBot  (Report bug) 08:46, 26 July 2017 (UTC)

Human interest in trying to inflate population numbers
How childish.. Nah, how human. Humans suck right..185.94.189.190 (talk) 19:57, 2 October 2017 (UTC)

title: Talk:Eindhoven


text:
Empty set

In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is 0|zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other theories, its existence can be deduced. Many possible properties of sets are vacuously true for the empty set.

Any set other than the empty set is called non-empty.

In some textbooks and popularizations, the empty set is referred to as the "null set". However, null set is a distinct notion within the context of measure theory, in which it describes a set of measure zero (which is not necessarily empty).

Notation
Common notations for the empty set include "{ }", "$$\emptyset$$", and "∅". The latter two symbols were introduced by the Bourbaki group (specifically André Weil) in 1939, inspired by the letter Ø in the Danish and Norwegian alphabets. In the past, "0" (the numeral zero) was occasionally used as a symbol for the empty set, but this is now considered to be an improper use of notation.

The symbol ∅ is available at Unicode point. It can be coded in HTML as ∅ and as ∅ or as ∅. It can be coded in LaTeX as \varnothing. The symbol $$\emptyset$$ is coded in LaTeX as \emptyset.

When writing in languages such as Danish and Norwegian, where the empty set character may be confused with the alphabetic letter Ø (as when using the symbol in linguistics), the Unicode character U+29B0 REVERSED EMPTY SET ⦰ may be used instead.

Properties
In standard axiomatic set theory, by the principle of extensionality, two sets are equal if they have the same elements (that is, neither of them has an element not in the other). As a result, there can be only one set with no elements, hence the usage of "the empty set" rather than "an empty set".

The only subset of the empty set is the empty set itself; equivalently, the power set of the empty set is the set containing only the empty set. The number of elements of the empty set (i.e., its cardinality) is zero. The empty set is the only set with either of these properties.

For any set A:
 * The empty set is a subset of A
 * The union of A with the empty set is A
 * The intersection of A with the empty set is the empty set
 * The Cartesian product of A and the empty set is the empty set

For any property P:
 * For every element of $$\varnothing$$, the property P holds (vacuous truth).
 * There is no element of $$\varnothing$$ for which the property P holds.

Conversely, if for some property P and some set V, the following two statements hold: then $$V = \varnothing.$$
 * For every element of V the property P holds
 * There is no element of V for which the property P holds

By the definition of subset, the empty set is a subset of any set A. That is, element x of $$\varnothing$$ belongs to A. Indeed, if it were not true that every element of $$\varnothing$$ is in A, then there would be at least one element of $$\varnothing$$ that is not present in A. Since there are elements of $$\varnothing$$ at all, there is no element of $$\varnothing$$ that is not in A. Any statement that begins "for every element of $$\varnothing$$" is not making any substantive claim; it is a vacuous truth. This is often paraphrased as "everything is true of the elements of the empty set."

In the usual set-theoretic definition of natural numbers, zero is modelled by the empty set.

Operations on the empty set
When speaking of the sum of the elements of a finite set, one is inevitably led to the convention that the sum of the elements of the empty set (the empty sum) is zero. The reason for this is that zero is the identity element for addition. Similarly, the product of the elements of the empty set (the empty product) should be considered to be one, since one is the identity element for multiplication.

A derangement is a permutation of a set without fixed points. The empty set can be considered a derangement of itself, because it has only one permutation ($$0!=1$$), and it is vacuously true that no element (of the empty set) can be found that retains its original position.

Extended real numbers
Since the empty set has no member when it is considered as a subset of any ordered set, every member of that set will be an upper bound and lower bound for the empty set. For example, when considered as a subset of the real numbers, with its usual ordering, represented by the real number line, every real number is both an upper and lower bound for the empty set. When considered as a subset of the extended reals formed by adding two "numbers" or "points" to the real numbers (namely negative infinity, denoted $$-\infty\!\,,$$ which is defined to be less than every other extended real number, and positive infinity, denoted $$+\infty\!\,,$$ which is defined to be greater than every other extended real number), we have that: $$\sup\varnothing=\min(\{-\infty, +\infty \} \cup \mathbb{R})=-\infty,$$ and $$\inf\varnothing=\max(\{-\infty, +\infty \} \cup \mathbb{R})=+\infty.$$

That is, the least upper bound (sup or supremum) of the empty set is negative infinity, while the greatest lower bound (inf or infimum) is positive infinity. By analogy with the above, in the domain of the extended reals, negative infinity is the identity element for the maximum and supremum operators, while positive infinity is the identity element for the minimum and infimum operators.

Topology
In any topological space X, the empty set is open by definition, as is X. Since the complement of an open set is closed and the empty set and X are complements of each other, the empty set is also closed, making it a clopen set. Moreover, the empty set is compact by the fact that every finite set is compact.

The closure of the empty set is empty. This is known as "preservation of nullary unions."

Category theory
If $$A$$ is a set, then there exists precisely one function $$f$$ from $$\varnothing$$ to $$A,$$ the empty function. As a result, the empty set is the unique initial object of the category of sets and functions.

The empty set can be turned into a topological space, called the empty space, in just one way: by defining the empty set to be open. This empty topological space is the unique initial object in the category of topological spaces with continuous maps. In fact, it is a strict initial object: only the empty set has a function to the empty set.

Set theory
In the von Neumann construction of the ordinals, 0 is defined as the empty set, and the successor of an ordinal is defined as $$S(\alpha)=\alpha\cup\{\alpha\}$$. Thus, we have $$0=\varnothing$$, $$1 = 0\cup\{0\}=\{\varnothing\}$$, $$2=1\cup\{1\}=\{\varnothing,\{\varnothing\}\}$$, and so on. The von Neumann construction, along with the axiom of infinity, which guarantees the existence of at least one infinite set, can be used to construct the set of natural numbers, $$\N_0$$, such that the Peano axioms of arithmetic are satisfied.

Historical issues
In the context of sets of real numbers, Cantor used $$P\equiv O$$ to denote "$$P$$ contains no single point". This $$\equiv O$$ notation was utilized in definitions; for example, Cantor defined two sets as being disjoint if their intersection has an absence of points; however, it is debatable whether Cantor viewed $$O$$ as an existent set on its own, or if Cantor merely used $$\equiv O$$ as an emptiness predicate. Zermelo accepted $$O$$ itself as a set, but considered it an "improper set".

Axiomatic set theory
In Zermelo set theory, the existence of the empty set is assured by the axiom of empty set, and its uniqueness follows from the axiom of extensionality. However, the axiom of empty set can be shown redundant in at least two ways:
 * Standard first-order logic implies, merely from the logical axioms, that exists, and in the language of set theory, that thing must be a set.  Now the existence of the empty set follows easily from the axiom of separation.
 * Even using free logic (which does not logically imply that something exists), there is already an axiom implying the existence of at least one set, namely the axiom of infinity.

Philosophical issues
While the empty set is a standard and widely accepted mathematical concept, it remains an ontological curiosity, whose meaning and usefulness are debated by philosophers and logicians.

The empty set is not the same thing as ; rather, it is a set with nothing it and a set is always. This issue can be overcome by viewing a set as a bag—an empty bag undoubtedly still exists. Darling (2004) explains that the empty set is not nothing, but rather "the set of all triangles with four sides, the set of all numbers that are bigger than nine but smaller than eight, and the set of all opening moves in chess that involve a king."

The popular syllogism
 * Nothing is better than eternal happiness; a ham sandwich is better than nothing; therefore, a ham sandwich is better than eternal happiness

is often used to demonstrate the philosophical relation between the concept of nothing and the empty set. Darling writes that the contrast can be seen by rewriting the statements "Nothing is better than eternal happiness" and "[A] ham sandwich is better than nothing" in a mathematical tone. According to Darling, the former is equivalent to "The set of all things that are better than eternal happiness is $$\varnothing$$" and the latter to "The set {ham sandwich} is better than the set $$\varnothing$$". The first compares elements of sets, while the second compares the sets themselves.

Jonathan Lowe argues that while the empty set
 * was undoubtedly an important landmark in the history of mathematics, … we should not assume that its utility in calculation is dependent upon its actually denoting some object.

it is also the case that:
 * "All that we are ever informed about the empty set is that it (1) is a set, (2) has no members, and (3) is unique amongst sets in having no members. However, there are very many things that 'have no members', in the set-theoretical sense—namely, all non-sets. It is perfectly clear why these things have no members, for they are not sets. What is unclear is how there can be, uniquely amongst sets, a which has no members. We cannot conjure such an entity into existence by mere stipulation."

George Boolos argued that much of what has been heretofore obtained by set theory can just as easily be obtained by plural quantification over individuals, without reifying sets as singular entities having other entities as members.

title: Empty set


text:
Egoism

Egoism is a philosophy concerned with the role of the self, or, as the motivation and goal of one's own action. Different theories of egoism encompass a range of disparate ideas and can generally be categorized into descriptive or normative forms. That is, they may be interested in either describing that people do act in self-interest or prescribing that they should. Other definitions of egoism may instead emphasise action according to one's will rather than one's self-interest, and furthermore posit that this is a truer sense of egoism.

The New Catholic Encyclopedia states of egoism that it "incorporates in itself certain basic truths: it is natural for man to love himself; he should moreover do so, since each one is ultimately responsible for himself; pleasure, the development of one's potentialities, and the acquisition of power are normally desirable." The moral censure of self-interest is a common subject of critique in egoist philosophy, with such judgments being examined as means of control and the result of power relations. Egoism may also reject that insight into one's internal motivation can arrive extrinsically, such as from psychology or sociology, though, for example, this is not present in the philosophy of Friedrich Nietzsche.

Overview
The term egoism is derived from the French, from the Latin (first person singular personal pronoun; "I") with the French  ("-ism").

Descriptive theories
The descriptive variants of egoism are concerned with self-regard as a factual description of human motivation and, in its furthest application, that all human motivation stems from the desires and interest of the ego. In these theories, action which is self-regarding may be simply termed egoistic.

The position that people tend to act in their own self-interest is called default egoism, whereas psychological egoism is the position that all motivations are rooted in an ultimately self-serving psyche. That is, in its strong form, that even seemingly altruistic actions are only disguised as such and are always self-serving. Its weaker form instead holds that, even if altruistic motivation is possible, the willed action necessarily becomes egoistic in serving one's own will. In contrast to this and philosophical egoism, biological egoism (also called evolutionary egoism) describes motivations rooted solely in reproductive self-interest (i.e. reproductive fitness). Furthermore, selfish gene theory holds that it is the self-interest of genetic information that conditions human behaviour.

Normative theories
Theories which hold egoism to be normative stipulate that the ego ought to promote its own interests above other values. Where this ought is held to be a pragmatic judgment it is termed rational egoism and where it is held to be a moral judgment it is termed ethical egoism. The Stanford Encyclopedia of Philosophy states that "ethical egoism might also apply to things other than acts, such as rules or character traits" but that such variants are uncommon. Furthermore, conditional egoism is a consequentialist form of ethical egoism which holds that egoism is morally right if it leads to morally acceptable ends. John F. Welsh, in his work Max Stirner's Dialectical Egoism: A New Interpretation, coins the term dialectical egoism to describe an interpretation of the egoist philosophy of Max Stirner as being fundamentally dialectical.

Normative egoism, as in the case of Stirner, need not reject that some modes of behavior are to be valued above others—such as Stirner's affirmation that non-restriction and autonomy are to be most highly valued. Contrary theories, however, may just as easily favour egoistic domination of others.

Nietzsche
The philosophy of Friedrich Nietzsche has been linked to forms of both descriptive and normative egoism. Nietzsche, in attacking the widely held moral abhorrence for egoistic action, seeks to free higher human beings from their belief that this morality is good for them. He rejects Christian and Kantian ethics as merely the disguised egoism of slave morality.

In his On the Genealogy of Morals, Friedrich Nietzsche traces the origins of master–slave morality to fundamentally egoistic value judgments. In the aristocratic valuation, excellence and virtue come as a form of superiority over the common masses, which the priestly valuation, in ressentiment of power, seeks to invert—where the powerless and pitiable become the moral ideal. This upholding of unegoistic actions is therefore seen as stemming from a desire to reject the superiority or excellency of others. He holds that all normative systems which operate in the role often associated with morality favor the interests of some people, often, though not necessarily, at the expense of others.

Nevertheless, Nietzsche also states in the same book that there is no 'doer' of any acts, be they selfish or not:

Jonas Monte of Brigham Young University argues that Nietzsche doubted if any 'I' existed in the first place, which the former defined as "a conscious Ego who commands mental states".

Other theoreticians

 * Jeremy Bentham, who is attributed as an early proponent of psychological egoism
 * Nikolai Gavrilovich Chernyshevskii, a Russian literary critic and philosopher of nihilism and rational egoism
 * Aleister Crowley, who popularized the expression "Do what thou wilt"
 * Arthur Desmond as Ragnar Redbeard (possibly, unproved)
 * Thomas Hobbes, who is attributed as an early proponent of psychological egoism
 * John Henry Mackay, a British-German egoist anarchist
 * Bernard de Mandeville, whose materialism has been retroactively described as form of egoism
 * Friedrich Nietzsche, whose concept of will to power has both descriptive and prescriptive interpretations
 * Dmitry Ivanovich Pisarev, a Russian literary critic and philosopher of nihilism and rational egoism
 * Ayn Rand, who supported an egoistic model of capitalist self-incentive and selfishness
 * Max Stirner, whose views were described by John F. Welsh as "dialectical egoism"
 * Benjamin Tucker, an American egoist anarchist
 * James L. Walker, who independently formulated an egoist philosophy before himself discovering the work of Stirner
 * John Fowles, British writer who laid out an individualist philosophy in his book The Aristos.

Relation to altruism
In 1851, French philosopher Auguste Comte coined the term altruism (altruisme;, ) as an antonym for egoism. In this sense, altruism defined Comte's position that all self-regard must be replaced with only the regard for others.

While Friedrich Nietzsche does not view altruism as a suitable antonym for egoism, Comte instead states that only two human motivations exist, egoistic and altruistic, and that the two cannot be mediated; that is, one must always predominate the other. For Comte, the total subordination of the self to altruism is a necessary condition to both social and personal benefit. Nietzsche, rather than rejecting the practice of altruism, warns that despite there being neither much altruism nor equality in the world, there is almost universal endorsement of their value and, notoriously, even by those who are its worst enemies in practice. Egoist philosophy commonly views the subordination of the self to altruism as either a form of domination that limits freedom, an unethical or irrational principle, or an extension of some egoistic root cause.

In evolutionary theory, biological altruism is the observed occurrence of an organism acting to the benefit of others at the cost of its own reproductive fitness. While biological egoism does grant that an organism may act to the benefit of others, it describes only such when in accordance with reproductive self-interest. Kin altruism and selfish gene theory are examples of this division. On biological altruism, the Stanford Encyclopedia of Philosophy states: "Contrary to what is often thought, an evolutionary approach to human behaviour does not imply that humans are likely to be motivated by self-interest alone. One strategy by which ‘selfish genes’ may increase their future representation is by causing humans to be non-selfish, in the psychological sense." This is a central topic within contemporary discourse of psychological egoism.

Relation to nihilism
The history of egoist thought has often overlapped with that of nihilism. For example, Max Stirner's rejection of absolutes and abstract concepts often places him among the first philosophical nihilists. The popular description of Stirner as a moral nihilist, however, may fail to encapsulate certain subtleties of his ethical thought. The Stanford Encyclopedia of Philosophy states, "Stirner is clearly committed to the non-nihilistic view that certain kinds of character and modes of behaviour (namely autonomous individuals and actions) are to be valued above all others. His conception of morality is, in this respect, a narrow one, and his rejection of the legitimacy of moral claims is not to be confused with a denial of the propriety of all normative or ethical judgement." Stirner's nihilism may instead be understood as cosmic nihilism. Likewise, both normative and descriptive theories of egoism further developed under Russian nihilism, shortly giving birth to rational egoism. Nihilist philosophers Dmitry Pisarev and Nikolay Chernyshevsky were influential in this regard, compounding such forms of egoism with hard determinism.

Max Stirner's philosophy strongly rejects modernity and is highly critical of the increasing dogmatism and oppressive social institutions that embody it. In order that it might be surpassed, egoist principles are upheld as a necessary advancement beyond the modern world. The Stanford Encyclopedia states that Stirner's historical analyses serve to "undermine historical narratives which portray the modern development of humankind as the progressive realisation of freedom, but also to support an account of individuals in the modern world as increasingly oppressed". This critique of humanist discourses especially has linked Stirner to more contemporary poststructuralist thought.

Political egoism
Since normative egoism rejects the moral obligation to subordinate the ego to society-at-large or a ruling class, it may be predisposed to certain political implications. The Internet Encyclopedia of Philosophy states: "Egoists ironically can be read as moral and political egalitarians glorifying the dignity of each and every person to pursue life as they see fit. Mistakes in securing the proper means and appropriate ends will be made by individuals, but if they are morally responsible for their actions they not only will bear the consequences but also the opportunity for adapting and learning."

In contrast with this however, such an ethic may not morally obligate against the egoistic exercise of power over others. On these grounds, Friedrich Nietzsche criticizes egalitarian morality and political projects as unconducive to the development of human excellence. Max Stirner's own conception, the union of egoists as detailed in his work The Ego and Its Own, saw a proposed form of societal relations whereby limitations on egoistic action are rejected. When posthumously adopted by the anarchist movement, this became the foundation for egoist anarchism.

Stirner's variant of property theory is similarly dialectical, where the concept of ownership is only that personal distinction made between what is one's property and what is not. Consequentially, it is the exercise of control over property which constitutes the nonabstract possession of it. In contrast to this, Ayn Rand incorporates capitalist property rights into her egoist theory.

Revolutionary politics
Egoist philosopher Nikolai Gavrilovich Chernyshevskii was the dominant intellectual figure behind the 1860–1917 revolutionary movement in Russia, which resulted in the assassination of Tsar Alexander II eight years before his death in 1889. Dmitry Pisarev was a similarly radical influence within the movement, though he did not personally advocate political revolution.

Philosophical egoism has also found wide appeal among anarchist revolutionaries and thinkers, such as John Henry Mackay, Benjamin Tucker, Émile Armand, Han Ryner Gérard de Lacaze-Duthiers, Renzo Novatore, Miguel Giménez Igualada, and Lev Chernyi. Though he did not involve in any revolutionary movements himself, the entire school of individualist anarchism owes much of its intellectual heritage to Max Stirner.

Egoist philosophy may be misrepresented as a principally revolutionary field of thought. However, neither Hobbesian nor Nietzschean theories of egoism approve of political revolution. Anarchism and revolutionary socialism were also strongly rejected by Ayn Rand and her followers.

Fascism
The philosophies of both Nietzsche and Stirner were heavily appropriated by fascist and proto-fascist ideologies. Nietzsche in particular has infamously been represented as a predecessor to Nazism and a substantial academic effort was necessary to disassociate his ideas from their aforementioned appropriation.

"At first sight, Nazi totalitarianism may seem the opposite of Stirner's radical individualism. But fascism was above all an attempt to dissolve the social ties created by history and replace them by artificial bonds among individuals who were expected to render explicit obedience to the state on grounds of absolute egoism. Fascist education combined the tenets of asocial egoism and unquestioning conformism, the latter being the means by which the individual secured his own niche in the system. Stirner's philosophy has nothing to say against conformism, it only objects to the Ego being subordinated to any higher principle: the egoist is free to adjust to the world if it is clear he will better himself by doing so. His 'rebellion' may take the form of utter servility if it will further his interest; what he must not do is to be bound by 'general' values or myths of humanity. The totalitarian ideal of a barrack-like society from which all real, historical ties have been eliminated is perfectly consistent with Stirner's principles: the egoist, by his very nature, must be prepared to fight under any flag that suits his convenience."

- Leszek Kołakowski

title: Egoism


text:
Endomorphism

In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space $m$ is a linear map $V$, and an endomorphism of a group $f: V → V$ is a group homomorphism $G$. In general, we can talk about endomorphisms in any category. In the category of sets, endomorphisms are functions from a set S to itself.

In any category, the composition of any two endomorphisms of $f: G → G$ is again an endomorphism of $X$. It follows that the set of all endomorphisms of $X$ forms a monoid, the full transformation monoid, and denoted $X$ (or $End(X)$ to emphasize the category $EndC(X)$).

Automorphisms
An invertible endomorphism of $C$ is called an automorphism. The set of all automorphisms is a subset of $X$ with a group structure, called the automorphism group of $End(X)$ and denoted $X$. In the following diagram, the arrows denote implication:

Endomorphism rings
Any two endomorphisms of an abelian group, $Aut(X)$, can be added together by the rule $A$. Under this addition, and with multiplication being defined as function composition, the endomorphisms of an abelian group form a ring (the endomorphism ring). For example, the set of endomorphisms of $$\mathbb{Z}^n$$ is the ring of all $(f + g)(a) = f(a) + g(a)$ matrices with integer entries. The endomorphisms of a vector space or module also form a ring, as do the endomorphisms of any object in a preadditive category. The endomorphisms of a nonabelian group generate an algebraic structure known as a near-ring. Every ring with one is the endomorphism ring of its regular module, and so is a subring of an endomorphism ring of an abelian group; however there are rings that are not the endomorphism ring of any abelian group.

Operator theory
In any concrete category, especially for vector spaces, endomorphisms are maps from a set into itself, and may be interpreted as unary operators on that set, acting on the elements, and allowing the notion of element orbits to be defined, etc.

Depending on the additional structure defined for the category at hand (topology, metric, ...), such operators can have properties like continuity, boundedness, and so on. More details should be found in the article about operator theory.

Endofunctions
An endofunction is a function whose domain is equal to its codomain. A homomorphic endofunction is an endomorphism.

Let $n × n$ be an arbitrary set. Among endofunctions on $S$ one finds permutations of $S$ and constant functions associating to every $S$ in $x$ the same element $S$ in $c$. Every permutation of $S$ has the codomain equal to its domain and is bijective and invertible. If $S$ has more than one element, a constant function on $S$ has an image that is a proper subset of its codomain, and thus is not bijective (and hence not invertible). The function associating to each natural number $S$ the floor of $n$ has its image equal to its codomain and is not invertible.

Finite endofunctions are equivalent to directed pseudoforests. For sets of size $n/2$ there are $n$ endofunctions on the set.

Particular examples of bijective endofunctions are the involutions; i.e., the functions coinciding with their inverses.

title: Endomorphism


text:
Talk:Eulers number

In The Book on Numbers by John Conway and Richard Guy the number e is persistently called Napier's number. I know that John Napier has more or less discovered logarithms, but is this really the correct name? --JanHidders

 * e is named after Euler. Whether that is correct or not. "Napier's number" gets very few google hits; compare and .--MarSch 30 June 2005 16:13 (UTC)

Delete this discussion
$e$ is a number, a fundamental object of mathematics. Each section of the article should be accessible to the widest, curious, audience possible. The nature of curiosity of this sort is not capricious. Hilbert's gravestone says it even better: "We can know. We will know." Mathematics is a word coined by Pythagorus meaning 'that which is learned' (according to The Mystery of the Aleph by Amir Aczel p.14.)

Specifically in my (now missing) case, Kronecker in is book On the Concept of Number attempted to banish from math what he regarded as unnecessary levels of abstraction, "anything... that could not be derived from integers is a finite number of steps" (according to John Derbyshire in Prime Obsession p. 185.) While recalling Kurt Gödel's proof that proof theory is deniable, and therefore that all of math is unprovable, I ask simply to play the game that it is, and to add to it.

Thus I wish to continue the now missing discussion I started concerning the short section Stochastic representations please. Where is the archive for the last talk page? Removing it was a careless mistake. The purpose of my missing discussion was to expand the article's audience. — Cp i r al Cpiral  21:27, 12 November 2011 (UTC)

title: Talk:Eulers number


text:
Eric Hoffer

Eric Hoffer (July 25, 1902 – May 21, 1983) was an American moral and social conservative philosopher. He was the author of ten books and was awarded the Presidential Medal of Freedom in February 1983. His first book, The True Believer (1951), was widely recognized as a classic, receiving critical acclaim from both scholars and laymen, although Hoffer believed that The Ordeal of Change (1963) was his finest work. The Eric Hoffer Book Award is an international literary prize established in his honor. The University of California, Berkeley awards an annual literary prize named jointly for Hoffer.

Early life
Many elements of Hoffer's early life are unverified, but in autobiographical statements, Hoffer claimed to have been born in 1902 in The Bronx, New York City, New York, to Knut and Elsa (Goebel) Hoffer. His parents were immigrants from Alsace, then part of Imperial Germany. By age five, Hoffer could already read in both English and his parents' native German. When he was five, his mother fell down the stairs with him in her arms. He later recalled, "I lost my sight at the age of seven. Two years before, my mother and I fell down a flight of stairs. She did not recover and died in that second year after the fall. I lost my sight and, for a time, my memory." Hoffer spoke with a pronounced German accent all his life, and spoke the language fluently. He was raised by a live-in relative or servant, a German immigrant named Martha. His eyesight inexplicably returned when he was 15. Fearing he might lose it again, he seized on the opportunity to read as much as he could. His recovery proved permanent, but Hoffer never abandoned his reading habit.

Hoffer was a young man when he also lost his father. The cabinetmaker's union paid for Knut Hoffer's funeral and gave Hoffer about $300 insurance money. He took a bus to Los Angeles and spent the next 10 years wandering, as he remembered, “up and down the land, dodging hunger and grieving over the world.” Hoffer eventually landed on Skid Row, reading, occasionally writing, and working at odd jobs.

In 1931, he considered suicide by drinking a solution of oxalic acid, but he could not bring himself to do it. He left Skid Row and became a migrant worker, following the harvests in California. He acquired a library card where he worked, dividing his time "between the books and the brothels." He also prospected for gold in the mountains. Snowed in for the winter, he read the Essays by Michel de Montaigne. Montaigne impressed Hoffer deeply, and Hoffer often made reference to him. He also developed a respect for America's underclass, which he said was "lumpy with talent."

Career
He wrote a novel, Four Years in Young Hank's Life, and a novella, Chance and Mr. Kunze, both partly autobiographical. He also penned a long article based on his experiences in a federal work camp, "Tramps and Pioneers." It was never published, but a truncated version appeared in Harper's Magazine after he became well known.

Hoffer tried to enlist in the U.S. Army at age 40 during World War II, but he was rejected due to a hernia. Instead, he began work as a longshoreman on the docks of San Francisco in 1943. At the same time, he began to write seriously.

Hoffer left the docks in 1964, and shortly after became an adjunct professor at the University of California, Berkeley. He later retired from public life in 1970. “I'm going to crawl back into my hole where I started,” he said. “I don't want to be a public person or anybody's spokesman... Any man can ride a train. Only a wise man knows when to get off.” In 1970, he endowed the Lili Fabilli and Eric Hoffer Laconic Essay Prize for students, faculty, and staff at the University of California, Berkeley.

Hoffer called himself an atheist but had sympathetic views of religion and described it as a positive force.

He died at his home in San Francisco in 1983 at the age of 80.

Working-class roots
Hoffer was influenced by his modest roots and working-class surroundings, seeing in it vast human potential. In a letter to Margaret Anderson in 1941, he wrote: "My writing is done in railroad yards while waiting for a freight, in the fields while waiting for a truck, and at noon after lunch. Towns are too distracting." He once remarked, "my writing grows out of my life just as a branch from a tree." When he was called an intellectual, he insisted that he simply was a longshoreman. Hoffer has been dubbed by some authors a "longshoreman philosopher."

Personal life
Hoffer, who was an only child, never married. He fathered a child with Lili Fabilli Osborne, named Eric Osborne, who was born in 1955 and raised by Lili Osborne and her husband, Selden Osborne. Lili Fabilli Osborne had become acquainted with Hoffer through her husband, a fellow longshoreman and acquaintance of Hoffer's. Despite this, Selden Osborne and Hoffer remained on good terms.

Hoffer referred to Eric Osborne as his son or godson. Lili Fabilli Osborne died in 2010 at the age of 93. Prior to her death, Osborne was the executor of Hoffer's estate, and vigorously controlled the rights to his intellectual property.

In his 2012 book Eric Hoffer: The Longshoreman Philosopher, journalist Tom Bethell revealed doubts about Hoffer's account of his early life. Although Hoffer claimed his parents were from Alsace-Lorraine, Hoffer himself spoke with a pronounced Bavarian accent. He claimed to have been born and raised in the Bronx but had no Bronx accent. His lover and executor Lili Fabilli stated that she always thought Hoffer was an immigrant. Her son, Eric Fabilli, said that Hoffer's life might have been comparable to that of B. Traven and considered hiring a genealogist to investigate Hoffer's early life, to which Hoffer reportedly replied, "Are you sure you want to know?" Pescadero land-owner Joe Gladstone, a family friend of the Fabilli's who also knew Hoffer, said of Hoffer's account of his early life: "I don't believe a word of it." To this day, no one ever has claimed to have known Hoffer in his youth, and no records apparently exist of his parents, nor indeed of Hoffer himself until he was about forty, when his name appeared in a census.

The True Believer
Hoffer came to public attention with the 1951 publication of his first book, The True Believer: Thoughts on the Nature of Mass Movements, which consists of a preface and 125 sections, which are divided into 18 chapters. Hoffer analyzes the phenomenon of "mass movements," a general term that he applies to revolutionary parties, nationalistic movements, and religious movements. He summarizes his thesis in §113: "A movement is pioneered by men of words, materialized by fanatics and consolidated by men of actions."

Hoffer argues that fanatical and extremist cultural movements, whether religious, social, or national, arise when large numbers of frustrated people, believing their own individual lives to be worthless or spoiled, join a movement demanding radical change. But the real attraction for this population is an escape from the self, not a realization of individual hopes: "A mass movement attracts and holds a following not because it can satisfy the desire for self-advancement, but because it can satisfy the passion for self-renunciation."

Hoffer consequently argues that the appeal of mass movements is interchangeable: in the Germany of the 1920s and the 1930s, for example, the Communists and National Socialists were ostensibly enemies, but sometimes enlisted each other's members, since they competed for the same kind of marginalized, angry, frustrated people. For the "true believer," Hoffer argues that particular beliefs are less important than escaping from the burden of the autonomous self.

Harvard historian Arthur M. Schlesinger Jr. said of The True Believer: "This brilliant and original inquiry into the nature of mass movements is a genuine contribution to our social thought."

Later works
Subsequent to the publication of The True Believer (1951), Eric Hoffer touched upon Asia and American interventionism in several of his essays. In "The Awakening of Asia" (1954), published in The Reporter and later his book The Ordeal of Change (1963), Hoffer discusses the reasons for unrest on the continent. In particular, he argues that the root cause of social discontent in Asia was not government corruption, "communist agitation," or the legacy of European colonial "oppression and exploitation," but rather that a "craving for pride" was the central problem in Asia, suggesting a problem that could not be relieved through typical American intervention.

During the Vietnam War, despite his objections to the antiwar movement and acceptance of the notion that the war was somehow necessary to prevent a third world war, Hoffer remained skeptical concerning American interventionism, specifically the intelligence with which the war was being conducted in Southeast Asia. After the United States became more involved in the war, Hoffer wished to avoid defeat in Vietnam because of his fear that such a defeat would transform American society for ill, opening the door to those who would preach a stab-in-the-back myth and allow for the rise of an American version of Hitler.

In The Temper of Our Time (1967), Hoffer implies that the United States as a rule should avoid interventions in the first place: "the better part of statesmanship might be to know clearly and precisely what not to do, and leave action to the improvisation of chance." In fact, Hoffer indicates that "it might be wise to wait for enemies to defeat themselves," as they might fall upon each other with the United States out of the picture. The view was somewhat borne out with the Cambodian-Vietnamese War and Chinese-Vietnamese War of the late 1970s.

Papers
Hoffer's papers, including 131 of the notebooks he carried in his pockets, were acquired in 2000 by the Hoover Institution Archives. The papers fill 75 ft of shelf space. Because Hoffer cultivated an aphoristic style, the unpublished notebooks (dated from 1949 to 1977) contain very significant work. Although available for scholarly study since at least 2003, little of their contents has been published. A selection of fifty aphorisms, focusing on the development of unrealized human talents through the creative process, appeared in the July 2005 issue of Harper's Magazine.

Published works

 * 1951 The True Believer: Thoughts On The Nature of Mass Movements. ISBN 0-06-050591-5
 * 1955 The Passionate State of Mind, and Other Aphorisms. ISBN 1-933435-09-7
 * 1963 The Ordeal of Change. ISBN 1-933435-10-0
 * 1967 The Temper of Our Time. ISBN 978-1-933435-22-0
 * 1968 Nature and The City
 * 1969 Working and Thinking on the Waterfront: A Journal, June 1958 to May 1959
 * 1971 First Things, Last Things
 * 1973 Reflections on the Human Condition. ISBN 1-933435-14-3
 * 1976 In Our Time
 * 1979 Before the Sabbath
 * 1982 Between the Devil and the Dragon: The Best Essays and Aphorisms of Eric Hoffer. ISBN 0-06-014984-1
 * 1983 Truth Imagined. ISBN 1-933435-01-1

Interviews

 * Conversations with Eric Hoffer, twelve-part television interview by James Day of KQED, San Francisco, 1963.
 * "Eric Hoffer: The Passionate State of Mind" with Eric Sevareid, CBS, September 19, 1967 (re-broadcast on November 14, due to popular demand).
 * "The Savage Heart: A Conversation with Eric Hoffer," with Eric Sevareid, CBS, January 28, 1969.

Awards and recognition

 * 1971, May – Honorary Doctorate; Stonehill College
 * 1971, June – Honorary Doctorate; Michigan Technological University
 * 1978 – Bust of Eric Hoffer by sculptor Jonathan Hirschfeld; commissioned by Charles Kittrell and placed in Bartlesville, Oklahoma
 * 1983, February 13 – Presidential Medal of Freedom awarded by Ronald Reagan
 * 1985, September 17 – Skygate unveiling in San Francisco; dedication speech by Eric Sevareid

title: Eric Hoffer