Question

Show that the relation R on the set A = {x ∈ Z ; 0 ≤ x ≤ 12}, given by R = {(a, b) : a = b}, is an equivalence relation. Find the set of all elements related to 1.

Answer 1

We observe the following properties of R.

Reflexivity: Let a be an arbitrary element of A. Then,
$$\begin{gathered} a\in R \\ \Rightarrow a=a\left[\mathrm{Since},\mathrm{every}\mathrm{element}\mathrm{is}\mathrm{equal}\mathrm{to}\mathrm{itself}\right] \\ \Rightarrow \left(a,a\right)\in R\mathrm{for}\mathrm{all}a\in A \\ \mathrm{So},R\mathrm{is}\mathrm{reflexive}\mathrm{on}A. \\ \mathrm{Symmetry}:\mathrm{Let}\left(a,b\right)\in R \\ \Rightarrow ab \\ \Rightarrow b=a \\ \Rightarrow \left(b,a\right)\in R\mathrm{for}\mathrm{all}a,b\in A \\ \mathrm{So},R\mathrm{is}\mathrm{symmetric}\mathrm{on}A. \\ \mathrm{Transitivity}:\mathrm{Let}\left(a,b\right)\mathrm{and}\left(b,c\right)\in R \\ \Rightarrow a=b\mathrm{and}b=c \\ \Rightarrow a=bc \\ \Rightarrow a=c \\ \Rightarrow \left(a,c\right)\in R \\ \mathrm{So},R\mathrm{is}\mathrm{transitive}\mathrm{on}A. \end{gathered}$$

Hence, R is an equivalence relation on A.

The set of all elements related to 1 is {1}.