# Let R be a relation on \mathbb{Z} defined by R=\{(p,q)\in\mathbb{Z}\times\mat

Let R be a relation on $$\displaystyle{\mathbb{{{Z}}}}$$ defined by
$$\displaystyle{R}={\left\lbrace{\left({p},{q}\right)}\in{\mathbb{{{Z}}}}\times{\mathbb{{{Z}}}}{\mid}{p}-{q}\right.}$$ is a multiple of $$\displaystyle{3}\rbrace$$
a) Show that R is reflexive.
b) Show that R is symmetric.
c) Show that R is transitive.

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BleabyinfibiaG

a) For any element $$\displaystyle{p}\in{\mathbb{{{Z}}}}$$
$$\displaystyle{p}-{p}={0}$$
is a multiple of 0.
Thus, $$\displaystyle{\left({p},{p}\right)}\in{R}\Rightarrow{R}$$ is reflexive.
b) Let $$\displaystyle{\left({p},{q}\right)}\in{R}$$ Then
$$\displaystyle{p}-{q}$$ is a multiple of 3, that is $$\displaystyle{p}-{q}={3}{k}$$ for some $$\displaystyle{k}\in{\mathbb{{{Z}}}}$$
To show that $$\displaystyle{\left({q},{p}\right)}\in{R}$$ Consider
$$\displaystyle{q}-{p}=-{\left({p}-{q}\right)}$$
$$\displaystyle=-{3}{k}={3}{m}$$
where $$\displaystyle{m}=-{k}\in{\mathbb{{{Z}}}}$$
So $$\displaystyle{q}-{p}$$ is a multiple of 3
$$\displaystyle\Rightarrow{\left({q},{p}\right)}\in{R}$$
$$\displaystyle\Rightarrow{R}$$ is symmetric
c) Let $$\displaystyle{\left({p},{q}\right)}\in{R}$$ and $$\displaystyle{\left({q},{r}\right)}\in{R}$$ Then both $$\displaystyle{p}-{q}$$ and $$\displaystyle{q}-{r}$$ are multiples of 3, that is,
$$\displaystyle{p}-{q}={3}{k},{k}\in{\mathbb{{{Z}}}}$$ and $$\displaystyle{q}-{r}={3}{m},{m}\in{R}$$
To show $$\displaystyle{\left({p},{r}\right)}\in{R}.$$ For that, consider
$$\displaystyle{p}-{r}={p}-{q}+{q}-{r}$$
$$\displaystyle={3}{k}+{3}{m}={3}{\left({k}+{m}\right)}$$
$$\displaystyle\Rightarrow{p}-{r}$$ is a multiple of 3.
$$\displaystyle\Rightarrow{\left({p},{r}\right)}\in{R}$$
Thus, R is transitive.