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

Let the relation $R$ on the set $M = \{1, 2, 3, \dots, 16\}$ be given by $R = \{(x, y) : 4y = 5x - 3, x, y \in M\}$. Then the minimum number of elements required to be added in $R$, in order to make the relation symmetric, is equal to

Let $R$ be the relation in the set $\{1, 2, 3\}$ given by $R = \{(1, 1), (2, 2), (3, 3)\}$. Choose the correct answer.

Let $N$ be the set of natural numbers and a relation $R$ on $N$ be defined by $R = \{(x, y) \in N \times N : x^{3}-3x^{2}y-xy^{2}+3y^{3}=0\}$. Then the relation $R$ is:

Show that the relation $R$ in the set $A$ of points in a plane given by $R = \{(P, Q) : \text{distance of the point } P \text{ from the origin is same as the distance of the point } Q \text{ from the origin}\}$,is an equivalence relation. Further,show that the set of all points related to a point $P \neq (0, 0)$ is the circle passing through $P$ with origin as centre.

Let $A = \{p, q, r\}$. Which of the following is an equivalence relation on $A$?