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I recently learned about equivalence relations and how equality between integers is an example of an equivalence relation. I also learned that operations such as addition (and possibly relations like equality) may be defined differently on different objects.

Consider the set of ordered pairs of natural numbers, $S=\mathbb{N}\times\mathbb{N}$. Then following is an equivalence relation on $S$:

Let $(a,b),(c,d)\in S$. Then $(a,b)R(c,d)$ if $a+d=b+c$.

I was wondering if we can define equality between elements of $S$ by the above relation. I suppose that we typically define equality on $S=\mathbb{N}\times\mathbb{N}$ as we do in $\mathbb{R}^2$:

Let $(a,b),(c,d)\in S$. Then $(a,b)=(c,d)$ if $a=c$ and $b=d$.

According to Wikipedia Definition of Equality, $x=y$ if and only if, given any predicate $P$, $P(x)$ if and only if $P(y)$. Would the following be an example to show that the definition of equality as $(a,b)=(c,d)$ if $a+d=b+c$ is invalid?

Under this definition of equality for elements of $S$, $(2,3)=(4,5)$. If $P$ is the predicate, $s\in S$ has first coordinate $2$, then $P(2,3)$ is true but $P(4,5)$ is false.

If so, would the only acceptable definition of equality in $S$ be $(a,b)=(c,d)$ if $a=c$ and $b=d$? This question Defining an Equality appears to be relevant but I am not sure how to interpret I.4.

Thank you for your help. I would appreciate an explanation suitable for a undergraduate like myself.

J. W. Tanner
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monkey king
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2 Answers2

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In the standard set theory of Zermelo-Fraenkel, two objects are equal if they are the same set, i.e. if all their elements are the same (axiom of extensionality).

This said, you can define different kind of equalities on sets -basing on your needs- through, for instance, equivalence relations. This does not violate the equality in ZF, it is just an internal definition (e.g. equivalence relation) that you can call "equality", but it is not the actual equality of sets.

What you are trying to define seems to be the set-theoretical definition of integers $\mathbb{Z}$. Integers are usually defined as $(\mathbb{N}\times \mathbb{N})/ R$, where $R$ is the equivalence relation on $S=\mathbb{N}\times \mathbb{N}$ that you defined.

The idea behind this definition is that a pair $(a,b)\in S$ in meant to represent $a-b$, thus $a-b=c-d \iff a+d=c+b$. The definition uses the sum instead of the difference because $\mathbb{N}$ is closed under the sum but not under difference, so using the sum $a+d=c+b$ allows us to define the relation without using the integers (which would be a problem since we are just defining them).

SilvioM
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  • Thank you for your help. I have heard of natural numbers being sets, e.g $0=\emptyset$, $1={\emptyset}={0}$, $2={\emptyset,{\emptyset}}={0,1}$, $3={\emptyset,{\emptyset,{\emptyset}}}={0,1,2}$, etc. So what set would, say, $(2,3)$ be? – monkey king Aug 22 '23 at 16:48
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    @monkeyking yes, this is the ZF set-theoretical definition of natural numbers by Von Neumann. While the set-theoretical definition of ordered pair (Kuratowski's definition) is: given two sets $A$ and $B$, an element $(a,b)\in A\times B$ is defined as ${ a, {a,b} }$. Thus, the element $(2,3)$ is actually ${ 2, { 2,3} }={ {1}, { {1}, {2} } }=...$. You can reconstruct the actual expression for $(2,3)$ by re-iterating the re-writing until you only use braces and $\emptyset$ – SilvioM Aug 22 '23 at 17:01
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Equality is a type of equivalence relation, but not the only type of equivalence relation. We say a relation is an equivalence relation if it has the same properties as equality (like reflexivity, symmetry, and transitivity). But many other equivalence relations say elements are equivalent if they share a certain property that doesn’t necessarily have to be equality. So yes, in your example, $(2,3)R(4,5)$, but that doesn’t mean $(2,3)=(4,5)$ because $=$ is a different equivalence relation than $R$. So yes, equality on $S$ holds if and only if the coordinates are equal, and that is one of the many equivalence relations you can have on $\mathbb{N}\times\mathbb{N}$.

Here are some common examples of equivalence relations:

  • In $\mathbb{Z}\times(\mathbb{Z}\setminus\{0\})$ we have $(a,b)R(c,d)$ if $ad=bc$, or in other words, if $a/b=c/d$. So in this example $(2,4)R(1,2)$ but we don’t have equality between these two points. In fact it’s just that the fractions are equivalent. This equivalence relation is used to construct the rational numbers, and (if you’ve learned about this yet) fields from integral domains.
  • In $\mathbb{Z}$ two elements are related if they are congruent modulo $n$, and this is an equivalence relation.
  • In group theory, if we have a group $G$, two elements are related if they are in the same coset of a subgroup $H$.

There are so many more examples of equivalence relations beyond equality that are helpful in a variety of ways in advanced mathematics. It will help to think more broadly about what equivalence means beyond elements being equal.

blakedylanmusic
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