Question

If R is a relation on the set A = {1, 2, 3} given by R = {(1, 1), (2, 2), (3, 3)}, then R is
(a) reflexive
(b) symmetric
(c) transitive
(d) all the three options

Answer 1

(d) all the three options

$$\begin{gathered} R=\left\{\left(a,b\right):a=b\mathrm{and}a,b\in A\right\} \\ \mathrm{Reflexivity}:\mathrm{Let}a\in A.\mathrm{Then}, \\ a=a \\ \Rightarrow \left(a,a\right)\in R\mathrm{for}\mathrm{all}a\in A \\ \mathrm{So},R\mathrm{is}\mathrm{reflexive}\mathrm{on}A. \\ \mathrm{Symmetry}:\mathrm{Let}a,b\in A\mathrm{such}\mathrm{that}\left(a,b\right)\in R.\mathrm{Then}, \\ \left(a,b\right)\in R \\ \Rightarrow a=b \\ \Rightarrow b=a \\ \Rightarrow \left(b,a\right)\in R\mathrm{for}\mathrm{all}a\in A \\ \mathrm{So},R\mathrm{is}\mathrm{symmetric}\mathrm{on}A. \\ \mathrm{Transitivity}:\mathrm{Let}a,b,c\in A\mathrm{such}\mathrm{that}\left(a,b\right)\in R\mathrm{and}\left(b,c\right)\in R.\mathrm{Then}, \\ \left(a,b\right)\in R\Rightarrow a=b \\ \mathrm{and}\left(b,c\right)\in R\Rightarrow b=c \\ \Rightarrow a=c \\ \Rightarrow \left(a,c\right)\in R\mathrm{for}\mathrm{all}a\in A \\ \mathrm{So},R\mathrm{is}\mathrm{transitive}\mathrm{on}A. \end{gathered}$$

Hence, R is an equivalence relation on A.