If A = {1, 2, 3, 4} define relations on A which have properties of being
Question:

If = {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.

Solution:

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

Relation $R$ satisfies reflexivity and transitivity.

$\Rightarrow(1,1),(2,2),(3,3) \in R$

and $(1,1),(2,1) \in R \Rightarrow(1,1) \in R$

However, $(2,1) \in R$, but $(1,2) \notin R$

(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.