If $A = \{ 1,2,3,4\}$, define relations on $A$ which have properties of being
(i) reflexive, transitive but not symmetric.
(ii) symmetric but neither reflexive nor transitive.
(iii) reflexive, symmetric and transitive.

It is given that,, $A = \{ 1,2,3,4\}$

(i) Let ${R_1} = \{ (1,1),(1,2),(2,3),(2,2),(1,3),(3,3)\}$

${R_1}$ is reflexive, since, (1,1)(2,2)(3,3) lie in ${R_1}$.

Now, $(1,2) \in {R_1},(2,3) \in {R_1} \Rightarrow (1,3) \in {R_1}$

Hence we can say that, ${R_1}$ is also transitive but (1,2)$\in {R_1} \Rightarrow (2,1) \notin {R_1}$.

So, it is not symmetric.

(ii) Let ${R_2} = \{ (1,2),(2,1)\}$

Now, $(1,2) \in {R_2},(2,1) \in {R_2}$

So, it is For symmetric.

(iii) Let ${R_3} = \{ (1,2),(2,1),(1,1),(2,2),(3,3),(1,3),(3,1),(2,3)\}$

Hence we can say that, ${R_3}$ is reflexive, symmetric and transitive.