The void relation on a set $A$ is

Let $R$ be a relation on $N$ defined by $x + 2y = 8$. The domain of $R$ is

Show that each of the relation $R$ in the set $A=\{x \in Z: 0 \leq x \leq 12\},$ given by $R =\{( a , b ): a = b \}$ is an equivalence relation. Find the set of all elements related to $1$ in each case.

Determine whether each of the following relations are reflexive, symmetric and transitive:

Relation $\mathrm{R}$ in the set $\mathrm{Z}$ of all integers defined as $\mathrm{R} =\{(\mathrm{x}, \mathrm{y}): \mathrm{x}-\mathrm{y}$ is an integer $\}$

The relation "congruence modulo $m$" is

Let $A$ be the non-void set of the children in a family. The relation $'x$ is a brother of $y'$ on $A$ is