In order that a relation $R$ defined on a non-empty set $A$ is an equivalence relation, it is sufficient, if $R$

If $A$ is the set of even natural numbers less than $8$ and $B$ is the set of prime numbers less than $7$, then the number of relations from $A$ to $B$ is

Let $R$ be the relation defined in the set $A=\{1,2,3,4,5,6,7\}$ by $R =\{(a, b):$  both $a$ and $b$ are either odd or even $\} .$ Show that $R$ is an equivalence relation. Further, show that all the elements of the subset $ \{1,3,5,7\}$ are related to each other and all the elements of the subset $\{2,4,6\}$ are related to each other, but no element of the subset $\{1,3,5,7\}$ is related to any element of the subset $\{2,4,6\} .$

The number of reflexive relations of a set with four elements is equal to

If $A = \{1, 2, 3\}$ , $B = \{1, 4, 6, 9\}$ and $R$ is a relation from $A$ to $B$ defined by ‘$x$ is greater than $y$’. The range of $R$ is

Let $R$ and $S$ be two relations on a set $A$. Then