Show that the relation $R$ in the set $Z$ of integers given by $R = \{(a, b) : 2 ext{ divides } (a - b)\}$ is an equivalence relation.

Question

Show that the relation $R$ in the set $Z$ of integers given by $R = \{(a, b) : 2 	ext{ divides } (a - b)\}$ is an equivalence relation.

Accepted Answer

> [!TLDR]
> Since $2\mid(a-b)$ is reflexive, symmetric and transitive on $\mathbb{Z}$, the relation $R$ is an equivalence relation.

Given $R = \{(a,b) : 2 	ext{ divides } (a-b)\}$ on the set $\mathbb{Z}$ of integers.

**Reflexive:** For any $a\in\mathbb{Z}$, $a - a = 0 = 2\cdot 0$, which is divisible by $2$. So $(a,a)\in R$ for all $a$. Hence $R$ is reflexive.

**Symmetric:** Let $(a,b)\in R$. Then $2\mid(a-b)$, i.e. $a-b = 2m$ for some integer $m$. Then $b - a = -2m = 2(-m)$, which is divisible by $2$. So $(b,a)\in R$. Hence $R$ is symmetric.

**Transitive:** Let $(a,b)\in R$ and $(b,c)\in R$. Then $a-b = 2m$ and $b-c = 2n$ for some integers $m, n$. Adding, …

Q.Show that the relation $R$ in the set $Z$ of integers given by $R = \{(a, b) : 2 \text{ divides } (a - b)\}$ is an equivalence relation.

Concept understanding — Equivalence Relation Proof

Proving a Relation is an Equivalence Relation

The three checks

A worked template

Q.Show that the relation RRR in the set ZZZ of integers given by R={(a,b):2 divides (a−b)}R = \{(a, b) : 2 \text{ divides } (a - b)\}R={(a,b):2 divides (a−b)} is an equivalence relation.

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Q.Show that the relation $R$ in the set $Z$ of integers given by $R = \{(a, b) : 2 \text{ divides } (a - b)\}$ is an equivalence relation.

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