Let $R$ be a relation on the set $\mathbb{R}$ of real numbers such that $aRb$ iff $a - b$ is an integer. Test whether $R$ is an equivalence relation. If so, find the equivalence class of $1$ and $\frac{1}{2}$.

Question

Accepted Answer

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> [!TLDR]
> $R$ is reflexive, symmetric and transitive, hence an equivalence relation; $[1]$ is the set of all integers and $\left[	frac12ight]$ is the set of all numbers of the form $	frac12+n,\ n\in\mathbb{Z}$.
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> $aRb \iff a-b\in\mathbb{Z}$ (an integer).
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> **Reflexive:** For any $a\in\mathbb{R}$, $a-a=0\in\mathbb{Z}$, so $aRa$. Reflexive. ✓
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> **Symmetric:** If $aRb$, then $a-b\in\mathbb{Z}$. Then $b-a = -(a-b)\in\mathbb{Z}$ too (negative of an integer is an integer), so $bRa$. Symmetric. ✓
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> **Transitive:** If $aRb$ and $bRc$, then $a-b\in\mathbb{Z}$ and $b-c\in\mathbb{Z}$. Then $a-c = (a-b)+(b-c)$ is a sum of two integers, hence an integer, so $aRc$. Transitive. ✓
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> Since $R$ is reflexive, symmetric and transitive, $R$ is an **equivalence relation**.
> …

Q.Let $R$ be a relation on the set $\mathbb{R}$ of real numbers such that $aRb$ iff $a - b$ is an integer. Test whether $R$ is an equivalence relation. If so, find the equivalence class of $1$ and $\frac{1}{2}$ .

Concept understanding — Equivalence Relation Proof

Proving a Relation is an Equivalence Relation

The three checks

A worked template

Q.Let RRR be a relation on the set R\mathbb{R}R of real numbers such that aRbaRbaRb iff a−ba - ba−b is an integer. Test whether RRR is an equivalence relation. If so, find the equivalence class of 111 and 12\frac{1}{2}21​.

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Q.Let $R$ be a relation on the set $\mathbb{R}$ of real numbers such that $aRb$ iff $a - b$ is an integer. Test whether $R$ is an equivalence relation. If so, find the equivalence class of $1$ and $\frac{1}{2}$ .

Unlock everything free for 14 days