Let $R = \{(1, 3), (4, 2), (2, 4), (2, 3), (3, 1)\}$ be a relation on the set $A = \{1, 2, 3, 4\}$. The relation $R$ is

$A$ relation $R$ defined on a set $A$ is anti-symmetric if $(a, b) \in R$ and $(b, a) \in R$ implies:

Show that the relation $R$ in the set $A$ of all the books in a library of a college,given by $R = \{(x, y) : x \text{ and } y \text{ have the same number of pages} \}$ is an equivalence relation.

The relation $R = \{(a, b) : \operatorname{gcd}(a, b) = 1, 2a \neq b, a, b \in \mathbb{Z}\}$ is:

Let $A = \{1, 2, 3\}$. The number of relations on $A$ containing $(1, 2)$ which are symmetric and transitive but not reflexive is . . . . . . .

Let $A = \{-2, -1, 0, 1, 2, 3, 4\}$. Let $R$ be a relation on $A$ defined by $xRy$ if and only if $2x + y \le 2$. Let $l$ be the number of elements in $R$. Let $m$ and $n$ be the minimum number of elements required to be added in $R$ to make it reflexive and symmetric relations respectively. Then $l + m + n$ is equal to: