Let a set $A=A_{1} \cup A_{2} \cup \ldots \cup A_{k,} \quad$ where $A_{ i } \cap A _{ j }=\phi$ for $i \neq j 1 \leq i , j \leq k$. Define the relation $R$ from $A$ to $A$ by $R=\left\{(x, y): y \in A_{i}\right.$ if and only if $\left.x \in A_{i}, 1 \leq i \leq k\right\}$. Then, $R$ is

Let $P$ be the relation defined on the set of all real numbers such that 

$P = \left\{ {\left( {a,b} \right):{{\sec }^2}\,a - {{\tan }^2}\,b = 1\,} \right\}$. Then $P$ is

Show that the relation $R$ in the set $A=\{1,2,3,4,5\}$ given by $R =\{(a, b):|a-b|$ 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

Show that the relation $R$ in the set $\{1,2,3\}$ given by $R =\{(1,2),(2,1)\}$ is symmetric but neither reflexive nor transitive.

Let $N$ be the set of natural numbers and a relation $R$ on $N$ be defined by $R=\left\{(x, y) \in N \times N: x^{3}-3 x^{2} y-x y^{2}+3 y^{3}=0\right\} .$ Then the relation $R$ is:

The number of relations, on the set $\{1,2,3\}$ containing $(1,2)$ and $(2,3)$, which are reflexive and transitive but not symmetric, is