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

Let $X$ be a non-void set. If $\rho_1$ and $\rho_2$ are transitive relations on $X$, then which of the following is true?

Show that the relation $R$ in the set $A=\{1,2,3,4,5\}$ given by $R =\{(a, b):|a-b| \text{ is even}\}$ is an equivalence relation. Show that all the elements of $\{1,3,5\}$ are related to each other and all the elements of $\{2,4\}$ are related to each other,but no element of $\{1,3,5\}$ is related to any element of $\{2,4\}$.

Give an example of a relation on a set $A = \{5, 6, 7\}$ which is symmetric but neither reflexive nor transitive.

Let $R_{1}$ and $R_{2}$ be two relations defined on the set of real numbers $\mathbb{R}$ by $a R_{1} b \iff ab \geq 0$ and $a R_{2} b \iff a \geq b$. Then:

Let $X$ be a family of sets and $R$ be a relation on $X$ defined by '$A$ is disjoint from $B$'. Then $R$ is