Test whether the relation $R=\{(m,n):3 ext{ divides }m-n\}$ on $\{1,2,3,\ldots,10\}$ is reflexive, symmetric or transitive. What is the conclusion?

Question

Test whether the relation $R=\{(m,n):3	ext{ divides }m-n\}$ on $\{1,2,3,\ldots,10\}$ is reflexive, symmetric or transitive. What is the conclusion?

Accepted Answer

>

> [!TLDR]
> $R=\{(m,n):3\mid(m-n)\}$ satisfies all three properties, so it is an equivalence relation.
>
> **Reflexivity:** for any $m$, $m-m=0$, and $3\mid0$. So $(m,m)\in R$ for all $m$ — **$R$ is reflexive**.
>
> **Symmetry:** if $(m,n)\in R$, then $3\mid(m-n)$, i.e. $m-n=3k$ for some integer $k$. Then $n-m=-3k=3(-k)$, so $3\mid(n-m)$, i.e. $(n,m)\in R$ — **$R$ is symmetric**.
>
> **Transitivity:** if $(m,n)\in R$ and $(n,p)\in R$, then $m-n=3k_1$ and $n-p=3k_2$ for integers $k_1,k_2$. Adding:
> $$(m-n)+(n-p)=m-p=3(k_1+k_2)$$
> so $3\mid(m-p)$, i.e. $(m,p)\in R$ — **$R$ is transitive**.
> …

Q.Test whether the relation $R=\{(m,n):3\text{ divides }m-n\}$ on $\{1,2,3,\ldots,10\}$ is reflexive, symmetric or transitive. What is the conclusion?

Concept understanding — Equivalence Relation Proof

Proving a Relation is an Equivalence Relation

The three checks

A worked template

Q.Test whether the relation R={(m,n):3 divides m−n}R=\{(m,n):3\text{ divides }m-n\}R={(m,n):3 divides m−n} on {1,2,3,…,10}\{1,2,3,\ldots,10\}{1,2,3,…,10} is reflexive, symmetric or transitive. What is the conclusion?

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Q.Test whether the relation $R=\{(m,n):3\text{ divides }m-n\}$ on $\{1,2,3,\ldots,10\}$ is reflexive, symmetric or transitive. What is the conclusion?

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