Give an example of a relation. Which is Symmetric and transitive but not reflexive.

If $A = \left\{ {x \in {z^ + }\,:x < 10} \right.$& and $x$ is a multiple of $3$ or $4\}$, where $z^+$ is the set of positive integers, then the total number of symmetric relations on $A$ is

Given the relation $R = \{(1, 2), (2, 3)\}$ on the set $A = {1, 2, 3}$, the minimum number of ordered pairs which when added to $R$ make it an equivalence relation is

Let $P = \{ (x,\,y)|{x^2} + {y^2} = 1,\,x,\,y \in R\} $. Then $P$ is

If $R_{1}$ and $R_{2}$ are equivalence relations in a set $A$, show that $R_{1} \cap R_{2}$ is also an equivalence relation.