Show that $R=\lbrace (a,b): 5\mid(a^2-b^2) \rbrace$ is an equivalence relation

Question

How can I show that this is an equivalence relation ?

$$R=\lbrace (a,b): 5\mid(a^2-b^2) \rbrace$$

Answer 1

An equivalence relation needs to be reflexive, symmetric and transitive.

• $aRa$
• $aRb \implies bRa$
• $aRb$ and $bRc \implies aRc$

So in this case, is it true that:

• $5|(a^2-a^2)$
• if $5|(a^2-b^2)$ then $5|(b^2-a^2)$
• if $5|(a^2-b^2)$ and $5|(b^2-c^2)$ then $5|(a^2-c^2)$

?

Answer 2

This is often a source of confusion among beginners, so I'll try to explain where the problem is.

Everybody knows intuitively what a relation among elements of a set is; examples from the real world are “being a parent of”, “being siblings”, “being taller” and so on.

However, defining what a relation is becomes circular, so in mathematics a more pragmatic approach is followed:

a relation is a set consisting of ordered pairs

If $R$ is a set of ordered pairs, or a relation, we say that $R$ is a relation on the set $A$ if, for all $(x,y)\in R$, we have $x\in A$ and $y\in A$.

For instance, the empty relation is a relation on every set. If $A$ is any set, the relation $\{(x,x):x\in A\}$ is a relation on $A$, usually called the identity on $A$.

A relation $R$ on a set $A$ is said

• reflexive when, for all $a\in A$, $(a,a)\in R$;
• symmetric when, for all $a,b\in A$, if $(a,b)\in R$, then $(b,a)\in R$;
• transitive when, for all $a,b,c\in A$, if $(a,b)\in R$ and $(b,c)\in R$, then $(a,c)\in R$.

The three properties together are shortened into saying that $R$ is an equivalence relation on $A$.

Where does the confusion start from? An alternative notation for $(a,b)\in R$ is $a\mathrel{R}b$. You should remember that you need to test two elements, when talking about an instance of the relation.

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Your relation is probably defined on the integers. So you have to answer the following questions:

1. Does $5\mid a^2-a^2$, for all $a\in\mathbb{Z}$? This gives “$(a,a)\in R$”.
2. Does $5\mid a^2-b^2$ imply $5\mid b^2-a^2$? This gives “if $(a,b)\in R$, then $(b,a)\in R$.
3. Do the conditions $5\mid a^2-b^2$ and $5\mid b^2-c^2$ imply $5\mid a^2-c^2$?