A relation from $P$ to $Q$ is

The relation $R =\{( a , b ): \operatorname{gcd}( a , b )=1,2 a \neq b , a , b \in Z \}$ is:

Show that the relation $R$ in the set $R$ of real numbers, defined as $R =\left\{(a, b): a \leq b^{2}\right\}$ is neither reflexive nor symmetric nor transitive.

If $R_{1}$ and $R_{2}$ are equivalence relations in a set $A$, show that $R_{1} \cap R_{2}$ is also an equivalence relation.

Let $R$ be a relation on $Z \times Z$ defined by$ (a, b)$$R(c, d)$ if and only if $ad - bc$ is divisible by $5$ . Then $\mathrm{R}$ is

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-