Let $R$ be a relation defined on $N$ such that $a R b$ if $2a + 3b$ is a multiple of $5$,where $a, b \in N$. Then $R$ is

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

Let $L$ be the set of all straight lines in a plane and the relation $R$ on $L$ is defined by $\alpha R \beta \Leftrightarrow \alpha \perp \beta$,where $\alpha, \beta \in L$. Then $R$ is:

Let $A = \{0, 3, 4, 6, 7, 8, 9, 10\}$ and $R$ be the relation defined on $A$ such that $R = \{(x, y) \in A \times A : x - y \text{ is an odd positive integer or } x - y = 2\}$. The minimum number of elements that must be added to the relation $R$ so that it becomes a symmetric relation is equal to $...........$.

If $A = \{1, 2, 3, \dots, m\}$,then the total number of reflexive relations that can be defined from $A \to A$ is: