Let $N$ be the set of natural numbers and the relation $R$ on $N \times N$ is defined by $(a, b) R (c, d)$ if $ad(b + c) = bc(a + d)$. Then $R$ is:

Let $R$ be a relation defined on the set $\{1,2,3,4\} \times \{1,2,3,4\}$ by $R = \{((a,b), (c,d)) : 2a + 3b = 3c + 4d\}$. Then the number of elements in $R$ is:

For any two real numbers $\theta$ and $\phi$, we define $\theta R \phi$ if and only if $\sec^{2} \theta - \tan^{2} \phi = 1$. The relation $R$ 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 $...........$.

$A$ relation $R$ on a non-empty set $A$ is an equivalence relation if $R$ is:

On the set of integers $Z$,a relation $S$ is defined as: $S = \{(x, y) \in Z \times Z : |x - y| < 1\}$. Which of the following is true about $S$?