Let $\mathrm{A}=\{1,2,3,4,5\}$. Let $\mathrm{R}$ be a relation on $\mathrm{A}$ defined by $x R y$ if and only if $4 x \leq 5 y$. Let $m$ be the number of elements in $\mathrm{R}$ and $\mathrm{n}$ be the minimum number of elements from $\mathrm{A} \times \mathrm{A}$ that are required to be added to $\mathrm{R}$ to make it a symmetric relation. Then $m+n$ is equal to:

Check whether the relation $R$ in $R$ defined by $S =\left\{(a, b): a \leq b^{3}\right\}$ is reflexive, symmetric or 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 $N \times N$ defined by $(a, b) R$ (c, d) if and only if $a d(b-c)=b c(a-d)$. Then $R$ is

Let the relations $R_1$ and $R_2$ on the set $\mathrm{X}=\{1,2,3, \ldots, 20\}$ be given by $\mathrm{R}_1=\{(\mathrm{x}, \mathrm{y}): 2 \mathrm{x}-3 \mathrm{y}=2\}$ and $\mathrm{R}_2=\{(\mathrm{x}, \mathrm{y}):-5 \mathrm{x}+4 \mathrm{y}=0\}$. If $\mathrm{M}$ and $\mathrm{N}$ be the minimum number of elements required to be added in $R_1$ and $R_2$, respectively, in order to make the relations symmetric, then $\mathrm{M}+\mathrm{N}$ equals

If $\mathrm{R}$ is the smallest equivalence relation on the set $\{1,2,3,4\}$ such that $\{(1,2),(1,3)\} \subset R$, then the number of elements in $\mathrm{R}$ is