The number of relations on the set $A = \{1, 2, 3\}$ containing at most $6$ elements including $(1, 2)$,which are reflexive and transitive but not symmetric,is . . . . . . .

Relation $S = \{(3,3), (4,4)\}$ on set $A = \{3, 4, 5\}$ is . . . . . . .

Determine whether the following relation is reflexive,symmetric,and transitive:
Relation $R$ in the set $A$ of human beings in a town at a particular time given by $R = \{(x, y): x \text{ and } y \text{ work at the same place}\}$.

Show that the relation $R$ in the set $Z$ of integers given by $R = \{(a, b) : 2 \text{ divides } a - b\}$ is an equivalence relation.

Let the relations $R_1$ and $R_2$ on the set $X = \{1, 2, 3, \ldots, 20\}$ be given by $R_1 = \{(x, y) : 2x - 3y = 2\}$ and $R_2 = \{(x, y) : -5x + 4y = 0\}$. If $M$ and $N$ are the minimum number of elements required to be added to $R_1$ and $R_2$,respectively,to make the relations symmetric,then $M + N$ equals

Let $A = \{1, 2, 3, 4, 5\}$. $A$ relation $R$ on $A$ is defined by $R = \{(x, y) | x, y \in A \text{ and } x < y\}$. Then $R$ is: