Talk:Equality (mathematics)

message: It is inadequate to characterize mathematical identity as equality of functions since mathematical logic recognizes equality not only of functions, which are N:1 relations, but of relations generally.  See Identity_of_indiscernibles#Identity_and_indiscernibility.  Jim Bowery (talk) 16:15, 27 September 2015 (UTC)
thread_title: Identity

message: As I understand it, each branch of mathematics defines equality independently. That is, the axioms of number theory (implicitly) define numeric equality as some particular equivalence relation over the set of integers, the axioms of set theory define set equality as some particular equivalence relation over the set of sets, etc.. But, aside from saying equality must always be an equivalence relation, mathematics seems to have nothing to say about equality "in general". Thus there seems to be no definition with which to make sense of the claim that, e.g. 3 = {1, 4, 9}. In addition, it seems I could define a class of mathematical objects without bothering to define any equality relation over them. Is any or all of this correct? If so, I have some changes in mind for the article. --Ryguasu 13:31, 9 Sep 2003 (EDT)
I think it fair to say (a) the = symbol is heavily overloaded in mathematical usage, and (b) typical manoeuvres such as passage to the quotient are carried out without regard to the existence of a computable equality relation.
Charles Matthews 09:05, 10 Sep 2003 (UTC)
thread_title: Equality in different branches of mathematics

message: is there a not equal to page or should there be a section on this page?  i was specifically looking for the different symbols, such as <> and != and ≠ - Omegatron 02:15, Jul 12, 2004 (UTC)
* ≠ redirects here now, so we should expand that section if there is anything to add. — brighterorange (talk) 15:00, 14 October 2005 (UTC)
thread_title:  not equal to 

message: Isn't it a circular definition to say that
1. equality is the only binary relation that is
   reflexive, transitive, symmetric and antisymmetric
and
2. a binary relation R is antisymmetric iff 
   R(x,y) and R(y,x) implies equality of x and y.
When it is possible to define antisymmetry without refering to equality, fine.- 193.175.133.66
:The first is not presented as a definition but as a property.--Patrick 23:47, 27 Aug 2004 (UTC)
thread_title:  circular definition of equality 

message: The article currently appears to have no references for verification.  While the facts are presumably correct, it will need published citations for the various pieces of information, such as to verify that the exact phrasing of various mathematical definitions are properly phrased according to independently published texts.  Dugwiki 22:05, 8 February 2007 (UTC)
thread_title:  Tagged as unreferenced 

message: Is it true that equality is antisymmetric? Doesn't symmetry mean that if A = B,
then B = A. Set inclusion is certainly antisymmetric: A C B, then B C A, in
general, is false. However, a definition of set equality would be if A C B
and B C A, then A = B. In this case, wouldn't we also say that B = A. If so,
doesn't that mean that set equality is symmetric?
Bkv2k (talk) 14:47, 1 November 2009 (UTC)
replies: text: The word antisymmetry doesn't mean "asymmetry." If you look at the definition, you'll see that antisymmetry holds trivially for equality.—PaulTanenbaum (talk) 16:40, 2 November 2009 (UTC)
thread_title:  Equality is Antisymmetric? 

message: From the article:
:::A stronger sense of equality is obtained if some form of Leibniz's law is added as an axiom; the assertion of this axiom rules out "bare particulars"—things that have all and only the same properties but are not equal to each other—which are possible in some logical formalisms.  The axiom states that two things are equal if they have all and only the same properties. Formally:
:::: Given any x and y, x = y if, given any predicate P, P(x) if and only if P(y).
Having a look at this definition, it is true for any x and y to be not equal and still have all their predicates identical - therefore Leibniz's law doesn't seem to be used. --Abdull (talk) 22:16, 4 September 2010 (UTC)
thread_title:  Definition using Leibniz's law? 

message: This article spends too much time discussing the identity relation, which logically is not what equality is -- it's a reification of equality, but equality must be defined axiomatically. Before you can define a relation, you need to define the properties of sets, and to do so, you need the ZF axioms. Those in turn presuppose a notion of equality, in particular the axiom of extensionality; without that, there is no single identity relation (it expresses the notion that "two objects are the same if they have the same properties", specifically the same elements).
In Peano arithmetic, there's a similar situation. There, = is a symbol is no defined meaning. It is taken as primary and its meaning becomes clear through the axioms that define when x = y. Qwertyus (talk) 12:51, 7 October 2012 (UTC)
thread_title:  Equality as a relation 

message: From the Logical definitions section:
: In this law, "P(x) if and only if P(y)" can be weakened to "P(x) if P(y)"; the modified law is equivalent to the original, since a statement that applies to "any x and y" applies just as well to "any y and x".
This argument doesn't seem valid to me: take x and y such that the set of predicates holding for y is a proper subset of the set of predicates holding for x (and therefore x ≠ y). Then the modified statement says that "x = y [which is false] if and only if, given any predicate P, P(x) if P(y) [which is true]", and therefore is false.
My question is then: do such x and y exist? In other words, do we consider "being equal to y" to be a predicate (which might give rise to a kind of chicken-and-egg issue)? If such is the case, then x and y as described above cannot exist: "x = y if y = y" would imply that x = y, a contradiction.
In any case, I feel like that remark can either be fixed or made clearer, but I'm not sure how.
Naim42 (talk) 14:48, 22 January 2020 (UTC)
replies: text: I agree, and deleted the paragraph. The succeeding one was wrong, too (see Identity_of_indiscernibles#Identity_and_indiscernibility). - Jochen Burghardt (talk) 19:05, 22 January 2020 (UTC)
thread_title:  Leibniz's law's weakened form 

message: A discussion is taking place to address the redirect :Distinct (mathematics). The discussion will occur at Wikipedia:Redirects for discussion/Log/2020 October 31#Distinct (mathematics) until a consensus is reached, and readers of this page are welcome to contribute to the discussion.  –Deacon Vorbis (carbon • videos) 17:28, 31 October 2020 (UTC)
thread_title:  "Distinct (mathematics)" listed at [[Wikipedia:Redirects for discussion|Redirects for discussion]] 

message: A discussion is taking place to address the redirect :≣. The discussion will occur at Wikipedia:Redirects for discussion/Log/2020 November 14#≣ until a consensus is reached, and readers of this page are welcome to contribute to the discussion.  𝟙𝟤𝟯𝟺𝐪𝑤𝒆𝓇𝟷𝟮𝟥𝟜𝓺𝔴𝕖𝖗𝟰 (𝗍𝗮𝘭𝙠) 12:09, 14 November 2020 (UTC)
thread_title:  "≣" listed at [[Wikipedia:Redirects for discussion|Redirects for discussion]] 

message: From the Equality_(mathematics)#Basic_properties section, the last sentence states symmetry and transitivity can be deduced from substitution and reflection. I think that's true if the substitution being discussed is "if a = b then (P(a) <=> P(b))". However, the substitution being discussed is "if a = b then F(a) = F(b)". I don't think symmetry and transitivity can be deduced from that version of substitution. — Preceding unsigned comment added by 2601:989:4401:3830::8763 (talk • contribs) 15:25, 17 July 2021 (UTC)
replies: text: If F(x)=(x=a), the substitution gives that if a=b, then (a=a)=(b=a). So, symmetry results from substitution and reflexivity. If F(x)=(x=c), then the substitution gives that if a=b, then (a=c)=(b=c). So, transitivity results from substitution alone. In these formulas, the = that appears inside parentheses must be viewed as a function that can return true or false. D.Lazard (talk) 14:05, 17 July 2021 (UTC)
thread_title:  Deriving symmetry and transitivity using substitution and reflection.