On the set $N$ of natural numbers,the relation $R$ is defined by $nRm$ if $n$ is a factor of $m$ (i.e.,$n|m$). Then $R$ is:

Let $T$ be the set of all triangles in a Euclidean plane and a relation $R$ on $T$ is defined as $aRb$ if and only if $a \sim b$ (where $a \sim b$ denotes that triangle $a$ is similar to triangle $b$) for all $a, b \in T$. Then $R$ is:

Let $R_{1}$ and $R_{2}$ be relations on the set $\{1, 2, \ldots, 50\}$ such that $R_{1} = \{(p, p^{n}) : p \text{ is a prime and } n \geq 0 \text{ is an integer}\}$ and $R_{2} = \{(p, p^{n}) : p \text{ is a prime and } n = 0 \text{ or } 1\}$. Then,the number of elements in $R_{1} - R_{2}$ is:

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

Let ${R_1}$ be a relation defined by ${R_1} = \{ (a, b) | a \ge b, a, b \in R \}$. Then ${R_1}$ is

On set $A = \{1, 2, 3\}$, relations $R$ and $S$ are given by $R = \{(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)\}$ and $S = \{(1, 1), (2, 2), (3, 3), (1, 3), (3, 1)\}$. Then,