Question

Let Z be the set of integers. Show that the relation
R = {(a, b) : a, b ∈ Z and a + b is even}
is an equivalence relation on Z.

Answer 1

We observe the following properties of R.

Reflexivity:
$$\begin{gathered} \mathrm{Let}a\mathrm{be}\mathrm{an}\mathrm{arbitrary}\mathrm{element}\mathrm{of}Z.\mathrm{Then}, \\ a\in R \\ \mathrm{Clearly},a+a=2a\mathrm{is}\mathrm{even}\mathrm{for}\mathrm{all}a\in Z. \\ \Rightarrow \left(a,a\right)\in R\mathrm{for}\mathrm{all}a\in Z \\ \mathrm{So},R\mathrm{is}\mathrm{reflexive}\mathrm{on}Z. \end{gathered}$$

Symmetry:
$$\begin{gathered} \mathrm{Let}\left(a,b\right)\in R \\ \Rightarrow a+b\mathrm{is}\mathrm{even} \\ \Rightarrow b+a\mathrm{is}\mathrm{even} \\ \Rightarrow \left(b,a\right)\in R\mathrm{for}\mathrm{all}a,b\in Z \\ \mathrm{So},R\mathrm{is}\mathrm{symmetric}\mathrm{on}Z. \end{gathered}$$

Transitivity:
$$\begin{gathered} \mathrm{Let}\left(a,b\right)\mathrm{and}\left(b,c\right)\in R \\ \Rightarrow a+b\mathrm{and}b+c\mathrm{are}\mathrm{even} \\ \mathrm{Now},\mathrm{let}a+b=2x\mathrm{for}\mathrm{some}x\in Z \\ \mathrm{and}b+c=2y\mathrm{for}\mathrm{some}y\in Z \\ \mathrm{Adding}\mathrm{the}\mathrm{above}\mathrm{two},\mathrm{we}\mathrm{get} \\ a+2b+c=2x+2y \\ \Rightarrow a+c=2(x+y-b),\mathrm{which}\mathrm{is}\mathrm{even}\mathrm{for}\mathrm{all}x,y,b\in Z \\ \mathrm{Thus},\left(a,c\right)\in R \\ \mathrm{So},R\mathrm{is}\mathrm{transitive}\mathrm{on}Z. \end{gathered}$$

Hence, R is an equivalence relation on Z.