# Talk:Equality (mathematics)

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Field:  Foundations, logic, and set theory

## Identity

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)

## Antisymmetry

Given the definition of "antisymmetric" on the binary relation page uses "=", isn't it a triviality to list it as a property of "="? -- Tarquin

Yes, it's pretty trivial that equality is antisymmetric. From a certain POV (the one that holds that equality is a purely logical relation with the axioms shown here), the other properties are trivial as well (since they follow from pure logic) -- but this one is particularly trivial (since it follows from pure logic even if you don't take that POV). Still, it's worth knowing that equality is unique among equivalence relations as the only antisymmetric one, so the fact has some use. -- Toby 08:10 Dec 1, 2002 (UTC)

## Equality in different branches of mathematics

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)

## not equal to

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)

## circular definition of equality

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)

## Japanese usage

I have removed the following text from the article:

```The equal sign that is currently used in Japan (・) also is used as a punctuation to separate
the first and last names when a western person's name is written in Katakana.
```

1. I cannot find any references for the use of ・ as an equality sign in Japan. The usual sign I see in Japanese texts is ＝.

2. Even if ・ has such a use, its relevance to a discussion of the symbol "=" is unclear, and the relevance of its regular use to the history of the aforementioned symbol is extremely unclear.

If someone can come up with a reference that describes or demonstrates this usage of ・ as an equal sign, it might be a good idea to add a section on "other equals signs around the world". (If there are any. Might Arabic or Sanskrit have their own?)

## ?

• "Instead of considering Leibniz's law as an axiom, it can also be taken as the definition of equality. The property of being an equivalence relation, as well as the properties given below, can then be proved: they become theorems." -- does that mean the axiom is equivalent to the definition?
• It should be noted somewhere in the article that equality as a logical notion is in a larger sense undefined. To go into philosophical discussions in an article on mathematics is always dangerous. Plus, many's the time that equivalence in one domain is used to define equality in another. For example, what's a fraction? --VKokielov 04:08, 25 October 2006 (UTC)
For the first question, it may depend on the further rules for equality. In the context of a concrete proof system, it may be the case that the system has a weak axiomatization for equality that does not allow to conclude from P(x) and x=y to P(y). Then the definition, which allows this inference, is stronger. But if you replace the first "if" in Leibniz's law by "if and only if", it is clearly equivalent; there is no logical difference between an axiom "P(x) iff Q(x)" and a definition "P(x) iff Q(x)". Also, the axiom of extensionality: f = g iff f(x) = g(x) for all x, as far as the "if" direction is concerned, is not an obvious consequence of Leibniz's law.
What exactly is the danger you refer to? The technical way to define the rational numbers is as equivalence classes as pairs of integers. Then equality of rational numbers is indeed plain equality, even if these classes are defined using equivalence in the base domain. All basic logical notions are "in a larger sense undefined", beacuse if we could define them, they would not be basic. Think of implication. Equality is in comparison very defined or at least definable.  --LambiamTalk 06:52, 25 October 2006 (UTC)
What bothers me is that this is all very technical, and people might be inclined to think that there's something magical about names of elements of a mathematical system, whereas it's not the names but the relationships between them that matter. --VKokielov 12:48, 25 October 2006 (UTC)
You speak like a category theorist! But I've removed the section on equality and isomorphism you wrote, because it was not really clear what it was trying to say, nor who the intended audience were. In my opinion it was too technical for the average reader, and not particularly informative for mathematicians. Personally I think the article, in particular the lead section, should be made more accessible to non-mathematicians, while the more technical stuff should be pushed to the back.  --LambiamTalk 14:55, 25 October 2006 (UTC)

## Tagged as unreferenced

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)

## Equality is Antisymmetric?

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)

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)

## Definition using Leibniz's law?

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)

## Equality as a relation

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)