Let $A = \{1, 2, 3\}$. Then the number of equivalence relations containing $(1, 2)$ is:

Let $Z$ be the set of all integers,$A = \{(x, y) \in Z \times Z : (x-2)^{2} + y^{2} \leq 4\}$,$B = \{(x, y) \in Z \times Z : x^{2} + y^{2} \leq 4\}$,and $C = \{(x, y) \in Z \times Z : (x-2)^{2} + (y-2)^{2} \leq 4\}$. If the total number of relations from $A \cap B$ to $A \cap C$ is $2^{p}$,then the value of $p$ is:

If $R$ is a relation on the set $N$ (set of natural numbers),defined by $R = \{(x, y) : 3x + 3y = 10\}$.
Statement-$1$: $R$ is symmetric.
Statement-$2$: $R$ is reflexive.
Statement-$3$: $R$ is transitive.
Determine the correct sequence of truth values for the given statements (where $T$ means true and $F$ means false).

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

$R = \{(1,1), (2,2), (3,3)\}$ is defined on the set $A = \{x : x \in N, x < 4\}$. Then the relation $R$ is . . . . . . .

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