The relation $S$ in the set $R$ of real numbers,defined as $S = \{(a, b) : a < b^2\}$ is a . . . . . . relation.

Let $L$ be the set of all lines in the $XY$ plane and $R$ be the relation in $L$ defined as $R = \{(L_1, L_2) : L_1 \text{ is parallel to } L_2\}$. Show that $R$ is an equivalence relation. Find the set of all lines related to the line $y = 2x + 4$.

Let $R$ be a relation from $Q$ to $Q$ defined by $R = \{(a, b) : a, b \in Q \text{ and } a - b \in Z \}$. Show that $(a, a) \in R$ for all $a \in Q$.

Give an example of a relation which is transitive but neither reflexive nor symmetric.

Let $R = \{(x,y) : x,y \in N \text{ and } x^2 - 4xy + 3y^2 = 0\}$,where $N$ is the set of all natural numbers. Then the relation $R$ is

The number of relations,defined on the set ${a, b, c, d}$,which are both reflexive and symmetric,is equal to: