Let $R$ be an equivalence relation defined on a set containing $6$ elements. The minimum number of ordered pairs that $R$ should contain is

Given a non-empty set $X$,consider $P(X)$ which is the set of all subsets of $X$. Define the relation $R$ in $P(X)$ as follows: For subsets $A, B$ in $P(X)$,$ARB$ if and only if $A \subset B$. Is $R$ an equivalence relation on $P(X)$? Justify your answer.

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:
$R = \{(x, y) : x \text{ is exactly } 7 \, cm \text{ taller than } y\}$

The minimum number of elements that must be added to the relation $R = \{(a, b), (b, c), (b, d)\}$ on the set $\{a, b, c, d\}$ so that it is an equivalence relation,is $.........$

Let $R = \{( P , Q ) \mid P \text{ and } Q \text{ are at the same distance from the origin} \}$ be a relation. Then the equivalence class of $(1, -1)$ is the set:

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