Let $R$ be a relation on $\mathbb{Z} \times \mathbb{Z}$ defined by $(a, b) R (c, d)$ if and only if $ad - bc$ is divisible by $5$. Then $R$ is

Let $A = \{1, 2, 3, \ldots, 20\}$. Let $R_1$ and $R_2$ be two relations on $A$ such that $R_1 = \{(a, b) : b \text{ is divisible by } a\}$ and $R_2 = \{(a, b) : a \text{ is an integral multiple of } b\}$. Then,the number of elements in $R_1 - R_2$ is equal to . . . . . . .

Let $N$ denote the set of all natural numbers. Define two binary relations on $N$ as $R_1 = \{(x,y) \in N \times N : 2x + y = 10\}$ and $R_2 = \{(x,y) \in N \times N : x + 2y = 10\}$. Then

If a relation $R$ on the set $\{1, 2, 3\}$ is defined by $R = \{(1, 1)\}$,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 $...........$.

Check whether the relation $R$ defined in the set $\{1, 2, 3, 4, 5, 6\}$ as $R = \{(a, b) : b = a + 1\}$ is reflexive,symmetric,or transitive.