$A$ relation $R$ on a non-empty set $A$ is an equivalence relation if $R$ is:

Let $R$ be a relation defined on the set $Z$ of all integers such that $x R y$ if and only if $x+2y$ is divisible by $3$. Then:

Let $R$ be a transitive relation on a set $A$ and $I$ be the identity relation on $A$. Then:

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

On the set of real numbers $R$, a relation $\rho$ is defined by $x \rho y$ if and only if $x-y$ is zero or an irrational number. Then:

Let the relations $R_1$ and $R_2$ on the set $X = \{1, 2, 3, \ldots, 20\}$ be given by $R_1 = \{(x, y) : 2x - 3y = 2\}$ and $R_2 = \{(x, y) : -5x + 4y = 0\}$. If $M$ and $N$ are the minimum number of elements required to be added to $R_1$ and $R_2$,respectively,to make the relations symmetric,then $M + N$ equals