Let $R = \{(a, a)\}$ be a relation on a set $A$. Then $R$ 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 relations $R$ from an $m$-element set $A$ to an $n$-element set $B$ satisfying the condition$\left(a, b_1\right) \in R,\left(a, b_2\right) \in R \Rightarrow b_1=b_2$ for $a \in A, b_1, b_2 \in B$ is

Let ${R_1}$ be a relation defined by ${R_1} = \{ (a,\,b)|a \ge b,\,a,\,b \in R\} $. Then ${R_1}$ is

Let $A =\{2,3,4,5, \ldots ., 30\}$ and $^{\prime} \simeq ^{\prime}$ be an equivalence relation on $A \times A ,$ defined by $(a, b) \simeq (c, d),$ if and only if $a d=b c .$ Then the number of ordered pairs which satisfy this equivalence relation with ordered pair $(4,3)$ is equal to :