Let $R_{1}$ and $R_{2}$ be relations on the set $\{1, 2, \ldots, 50\}$ such that $R_{1} = \{(p, p^{n}) : p \text{ is a prime and } n \geq 0 \text{ is an integer}\}$ and $R_{2} = \{(p, p^{n}) : p \text{ is a prime and } n = 0 \text{ or } 1\}$. Then,the number of elements in $R_{1} - R_{2}$ is:

Let $N$ be the set of natural numbers greater than $100$. Define the relation $R$ on $N$ by: $R = \{(x, y) \in N \times N : \text{the numbers } x \text{ and } y \text{ have at least two common divisors}\}.$ Then $R$ is-

The relation $R$ defined on a set $A$ is antisymmetric if $(a, b) \in R$ and $(b, a) \in R$ implies $a = b$ for all $a, b \in A$. Based on this definition,the relation $R$ is antisymmetric if $(a, b) \in R$ and $(b, a) \in R$ implies $a = b$,which is equivalent to saying that if $a \neq b$,then it is not possible for both $(a, b) \in R$ and $(b, a) \in R$ to be true. Therefore,the condition is that for $a \neq b$,we cannot have both $(a, b) \in R$ and $(b, a) \in R$.

Let $M$ denote the 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 the relation $R_{1}$ be defined on $R$ as $a R_{1} b$ if $1+ab > 0$. Then

Let $N$ be the set of natural numbers and a relation $R$ on $N$ be defined by $R = \{(x, y) \in N \times N : x^{3}-3x^{2}y-xy^{2}+3y^{3}=0\}$. Then the relation $R$ is: