Question

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.

Answer 1

(i) The relation on A having properties of being reflexive, transitive, but not symmetric is
R = {(1, 1), (2, 2), (3, 3), (4, 4), (2, 1)}

$$\begin{gathered} \mathrm{Relation}R\mathrm{satisfies}\mathrm{reflexivity}\mathrm{and}\mathrm{transitivity}. \\ \Rightarrow \left(1,1\right),\left(2,2\right),\left(3,3\right)\in R \\ \mathrm{and}\left(1,1\right),\left(2,1\right)\in R\Rightarrow \left(1,1\right)\in R \\ \mathrm{However},\left(2,1\right)\in R,\mathrm{but}\left(1,2\right)\notin R \end{gathered}$$

(ii) The relation on A having properties of being symmetric, but neither reflexive nor transitive is
R = {(1, 2), (2, 1)}
The relation R on A is neither reflexive nor transitive, but symmetric.

(iii) The relation on A having properties of being symmetric, reflexive and transitive is
R = {(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (2, 1)}
The relation R is an equivalence relation on A.