The number of symmetric relations defined on the set $\{1, 2, 3, 4\}$ which are not reflexive is

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:

Show that the number of equivalence relations on the set $A = \{1, 2, 3\}$ containing $(1, 2)$ and $(2, 1)$ is $2$.

Let $A$ be the set of all students of a boys' school. Show that the relation $R$ in $A$ given by $R = \{(a, b) : a \text{ is sister of } b\}$ is the empty relation and $R^{\prime} = \{(a, b) : \text{the difference between heights of } a \text{ and } b \text{ is less than } 3 \text{ meters}\}$ is the universal relation.

Show that the relation $R$ in the set $A=\{1,2,3,4,5\}$ given by $R =\{(a, b):|a-b| \text{ is even}\}$ is an equivalence relation. Show that all the elements of $\{1,3,5\}$ are related to each other and all the elements of $\{2,4\}$ are related to each other,but no element of $\{1,3,5\}$ is related to any element of $\{2,4\}$.

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