Question

Show that the relation R defined by R = {(a, b) : a – b is divisible by 3; a, b ∈ Z} is an equivalence relation.

Answer 1

We observe the following relations of relation R.

Reflexivity:
$$\begin{gathered} \mathrm{Let}a\mathrm{be}\mathrm{an}\mathrm{arbitrary}\mathrm{element}\mathrm{of}R.\mathrm{Then}, \\ a-a=0=0\times 3 \\ \Rightarrow a-a\mathrm{is}\mathrm{divisible}\mathrm{by}3 \\ \Rightarrow \left(a,a\right)\in R\mathrm{for}\mathrm{all}a\in Z \\ \mathrm{So},R\mathrm{is}\mathrm{reflexive}\mathrm{on}Z. \end{gathered}$$

Symmetry:
$$\begin{gathered} \mathrm{Let}\left(a,b\right)\in R \\ \Rightarrow a-b\mathrm{is}\mathrm{divisible}\mathrm{by}3 \\ \Rightarrow a-b3p\mathrm{for}\mathrm{some}p\in Z \\ \Rightarrow b-a=3\left(-p\right) \\ \mathrm{Here},-p\in Z \\ \Rightarrow b-a\mathrm{is}\mathrm{divisible}\mathrm{by}3 \\ \Rightarrow \left(b,a\right)\in R\mathrm{for}\mathrm{all}a,b\in Z \\ \mathrm{So,}\mathit{R}\mathrm{is}\mathrm{symmetric}\mathrm{on}Z. \end{gathered}$$

Transitivity:
$$\begin{gathered} \mathrm{Let}\left(a,b\right)\mathrm{and}\left(b,c\right)\in R \\ \Rightarrow a-b\mathrm{and}b-c\mathrm{are}\mathrm{divisible}\mathrm{by}3 \\ \Rightarrow a-b=3p\mathrm{for}\mathrm{some}p\in Z \\ \mathrm{and}b-c=3q\mathrm{for}\mathrm{some}q\in Z \\ \mathrm{Adding}\mathrm{the}\mathrm{above}\mathrm{two},\mathrm{we}\mathrm{get} \\ a-b+b-c=3p+3q \\ \Rightarrow a-c=3\left(p+q\right) \\ \mathrm{Here},p+q\in Z \\ \Rightarrow a-c\mathrm{is}\mathrm{divisible}\mathrm{by}3 \\ \Rightarrow \left(a,c\right)\in R\mathrm{for}\mathrm{all}a,c\in Z \\ \mathrm{So},R\mathrm{is}\mathrm{transitive}\mathrm{on}Z. \end{gathered}$$

Hence, R is an equivalence relation on Z.