Let a relation $R$ on $\mathbb{N} \times \mathbb{N}$ be defined as : $\left(\mathrm{x}_1, \mathrm{y}_1\right) \mathrm{R}\left(\mathrm{x}_2, \mathrm{y}_2\right)$ if and only if $\mathrm{x}_1 \leq \mathrm{x}_2$ or $\mathrm{y}_1 \leq \mathrm{y}_2$

Consider the two statements :

($I$) $\mathrm{R}$ is reflexive but not symmetric.

($II$) $\mathrm{R}$ is transitive

Then which one of the following is true?

If $R$ be a relation $<$ from $A = \{1,2, 3, 4\}$ to $B = \{1, 3, 5\}$ i.e., $(a,\,b) \in R \Leftrightarrow a < b,$ then $Ro{R^{ – 1}}$ is

Given a non empty set $X$, consider $P ( X )$ which is the set of all subsets of $X$.

Define the relation $R$ in $P(X)$ as follows :

For subsets $A,\, B$ in $P(X),$ $ARB$ if and only if $A \subset B .$ Is $R$ an equivalence relation on $P ( X ) ?$ Justify your answer.

Let $N$ be the set of natural numbers greater than $100. $ Define the relation $R$ by : $R = \{(x,y) \in \,N × N :$ the numbers $x$ and $y$ have atleast two common divisors$\}.$ Then $R$ 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 a multiple of $4\}$