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

Let $M$ denotes set of all $3 \times 3$ non singular matrices. Define the relation $R$ by

$R = \{ (A,B) \in M \times M$ : $AB = BA\} ,$ 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.

A relation on the set $A\, = \,\{ x\,:\,\left| x \right|\, < \,3,\,x\, \in Z\} ,$ where $Z$ is the set of integers is defined by $R= \{(x, y) : y = \left| x \right|, x \ne  - 1\}$. Then the number of elements in the power set of $R$ is

Let $r$ be a relation from $R$ (set of real numbers) to $R$ defined by $r = \{(a,b) \, | a,b \in R$  and  $a - b + \sqrt 3$ is an irrational number$\}$ The relation $r$ is

The number of symmetric relations defined on the set $\{1,2,3,4\}$ which are not reflexive is