Question

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

Answer 1

(i) we define a relation $R_1$ as
$R_1$$=\left\{\right(1,1),(2,2),(3,3),(4,4),(1,2),(2,3),(1,3\left)\right\}$
Then it is easy to check that $R_1$ is reflexive, transitive but not symmetric. Students are advised to write other relations of this type.
(ii) Define $R_2$ as: $R_2$ $=\left\{\right(1,2),(2,1\left)\right\}$
Ti is clear that $R_2$ is symmetric but neither reflexive nor transitive. Write other relations of this type.
(iii) We define $r_3$ as follows:
$R_3$ $=\left\{\right(1,1),(2,2),(3,3),(4,4),(1,2),(2,1\left)\right\}$.
Then evidently $R_3$ is reflexive, symmetric and transitive, that is, $R_3$ is an equivalence relation on A.
(1, 2) $\in R_3,(2,1)\in R_3\Rightarrow (1,1)\in R_3$