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

Let $S$ be set of all real numbers ; then on set $S$ relation $R$ defined as $R = \{\ (a, b) : 1 + ab > 0\ \}$ is

Among the relations $S =\left\{( a , b ): a , b \in R -\{0\}, 2+\frac{ a }{ b } > 0\right\}$ And $T =\left\{( a , b ): a , b \in R , a ^2- b ^2 \in Z \right\}$,

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 $\{2,4,6\} .$

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

Give an example of a relation. Which is Reflexive and transitive but not symmetric.