Show that the relation R in the set A = {1, 2, 3, 4, 5} given by R = {(a, b) : |a − b| is even}, is an equivalence relation.

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> [!TLDR]
> $R$ is an equivalence relation because $|a-b|$ being even is exactly the condition that $a$ and $b$ have the same parity — a property that is reflexive, symmetric and transitive.
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> **Concept.** $R$ on a set $A$ is an equivalence relation if it is **reflexive** ($aRa$), **symmetric** ($aRb\Rightarrow bRa$) and **transitive** ($aRb,\,bRc\Rightarrow aRc$).
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> **Step-by-step.** Here $A=\{1,2,3,4,5\}$ and $aRb\iff|a-b|$ is even (equivalently, $a$ and $b$ are of the same parity).
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> - **Reflexive:** For any $a\in A$, $|a-a|=0$, which is even, so $aRa$. Reflexive. ✔
> - **Symmetric:** If $aRb$ then $|a-b|$ is even. Since $|b-a|=|a-b|$, $|b-a|$ is also even, so $bRa$. Symmetric. ✔ …

Q.Show that the relation R in the set A = {1, 2, 3, 4, 5} given by R = {(a, b) : |a − b| is even}, is an equivalence relation.

Concept understanding — Equivalence Relation Proof

Proving a Relation is an Equivalence Relation

The three checks

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Q.Show that the relation R in the set A = {1, 2, 3, 4, 5} given by R = {(a, b) : |a − b| is even}, is an equivalence relation.

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