Question

Let R be reflexive and transitive relation on a set S. Then R intersect R inverse is a
a)reflexive but not transitive relation.
b)transitive but not reflexive relation.
c)symmetric but not reflexive and transitive relation.
d)equivalence relation.

Answer 1

$$R = \left\{(a, b) \mid a R b \right\}$$ $$\forall (a, a): a R a$$ $$a R b \wedge b R c \Rightarrow a R c$$ $$R ^ {\prime} = \left\{(a, b) \mid b R a \right\}$$ $$a R ^ {\prime} b \equiv b R a$$ $$a R a \equiv a R ^ {\prime} a$$ $$a R ^ {\prime} b \wedge b R ^ {\prime} c \Rightarrow b R a \wedge c R b \Rightarrow c R a \equiv a R ^ {\prime} c$$

So, $R'$ is again reflexive and transitive

Intersection of two reflexive and transitive relations is again reflexive and transitive relation.

If $(a,b)\in R\cap R^{\prime}$ then:

$$a R b \equiv b R ^ {\prime} a \Rightarrow (b, a) \in R ^ {\prime}$$ $$a R ^ {\prime} b \equiv b R a \Rightarrow \Rightarrow (b, a) \in R$$ $$(a, b) \in R \cap R ^ {\prime} \Rightarrow (b, a) \in R \cap R ^ {\prime}$$

Thus, intersection of reflexive and transitive relation and its inverse is **equivalence relation**.