Let $A$ be a non-empty set of children in a family. The relation $R$ is defined on $A$ as '$x$ is a brother of $y$'. This relation is:

Let the set of all relations $R$ on the set $\{a, b, c, d, e, f\}$ be denoted by $S$,such that $R$ is reflexive and symmetric,and $R$ contains exactly $10$ elements. Then the number of elements in $S$ is $...$ .

Let $N$ denote the set of all natural numbers. Define two binary relations on $N$ as $R_1 = \{(x,y) \in N \times N : 2x + y = 10\}$ and $R_2 = \{(x,y) \in N \times N : x + 2y = 10\}$. Then

Show that 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 $1$.

Let $R$ and $S$ be two relations on a set $A$. Then which of the following is true?

Determine whether the following relation is reflexive,symmetric,and transitive:
Relation $R$ in the set $A = \{1, 2, 3, \ldots, 13, 14\}$ defined as $R = \{(x, y) : 3x - y = 0\}$.