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

Let $A = \{2, 4, 6, 8\}$. $A$ relation $R$ on $A$ is defined by $R = \{(2, 4), (4, 2), (4, 6), (6, 4)\}$. Then $R$ 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 . . . . . . .

Let $P(S)$ denote the power set of $S = \{1, 2, 3, \ldots, 10\}$. Define the relations $R_1$ and $R_2$ on $P(S)$ as $A R_1 B$ if $(A \cap B^c) \cup (B \cap A^c) = \varnothing$ and $A R_2 B$ if $A \cup B^c = B \cup A^c, \forall A, B \in P(S)$. Then:

Define a relation $R$ over a class of $n \times n$ real matrices $A$ and $B$ as $A R B$ if and only if there exists a non-singular matrix $P$ such that $P A P^{-1} = B$. Then which of the following is true?

Let $\rho$ be a relation defined on $N$, the set of natural numbers, as $\rho = \{(x, y) \in N \times N : 2x + y = 41\}$. Then: