Let $R$ be a relation over the set $N × N$ and it is defined by $(a,\,b)R(c,\,d) \Rightarrow a + d = b + c.$ Then $R$ is

Let $R$ be a relation on the set $A$ of ordered pairs of positive integers defined by $(x, y) R (u, v)$ if and only if $x v=y u .$ Show that $R$ is an equivalence relation.

The relation $R$ defined on a set $A$ is antisymmetric if $(a,\,b) \in R \Rightarrow (b,\,a) \in R$ for

Let $A=\{1,2,3, \ldots 20\}$. Let $R_1$ and $R_2$ two relation on $\mathrm{A}$ such that $\mathrm{R}_1=\{(\mathrm{a}, \mathrm{b}): \mathrm{b}$ is divisible by $\mathrm{a}\}$ $\mathrm{R}_2=\{(\mathrm{a}, \mathrm{b}): \mathrm{a}$ is an integral multiple of $\mathrm{b}\}$. Then, number of elements in $R_1-R_2$ is equal to_____.

Let $R = \{(1, 3), (2, 2), (3, 2)\}$ and $S = \{(2, 1), (3, 2), (2, 3)\}$ be two relations on set $A = \{1, 2, 3\}$. Then $RoS =$