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:

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 an equivalence relation. Find the set of all elements related to $1$ in each case.

Let $A = \{1, 2, 3, 4\}$ and $R$ be a relation in $A$ given by $R = \{(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (2, 1), (3, 1), (1, 3)\}$. Then $R$ is

Let $S$ be set of all real numbers ; then on set $S$ relation $R$ defined as $R = \{\ (a, b) : 1 + ab > 0\ \}$ is

An integer $m$ is said to be related to another integer $n$ if $m$ is a multiple of $n$. Then the relation is

Let $A=\{2,3,6,7\}$ and $B=\{4,5,6,8\}$. Let $R$ be a relation defined on A $\times$ B by $\left(a_1, b_1\right) R\left(a_2, b_2\right)$ is and only if $a_1+a_2=b_1+b_2$. Then the number of elements in $\mathrm{R}$ is ...........