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

Let $R$ be a relation on $N \times N$ defined by $(a, b) R (c, d)$ if and only if $ad(b-c) = bc(a-d)$. Then $R$ is

Let $L$ be the set of all lines in the $XY$ plane and $R$ be the relation in $L$ defined as $R = \{(L_1, L_2) : L_1 \text{ is parallel to } L_2\}$. Show that $R$ is an equivalence relation. Find the set of all lines related to the line $y = 2x + 4$.

On the set $R$ of real numbers, we define $x P y$ if and only if $x y \geq 0$. Then, the relation $P$ is

Let $A = \{2, 3, 5, 7, 9\}$. Let $R$ be the relation on $A$ defined by $xRy$ if and only if $2x \le 3y$. Let $l$ be the number of elements in $R$, and $m$ be the minimum number of elements required to be added in $R$ to make it a symmetric relation. Then $l + m$ is equal to:

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