Let $\mathrm{A}$ be the set of all students of a boys school. Show that the relation $\mathrm{R}$ in A given by $\mathrm{R} =\{(a, b): \mathrm{a} $ is sister of $\mathrm{b}\}$ is the empty relation and $\mathrm{R} ^{\prime}=\{(a, b)$ $:$ the difference between heights of $\mathrm{a}$ and $\mathrm{b}$ is less than $3\,\mathrm{meters}$ $\}$ is the universal relation. 

Let $A =\{2,3,4,5, \ldots ., 30\}$ and $^{\prime} \simeq ^{\prime}$ be an equivalence relation on $A \times A ,$ defined by $(a, b) \simeq (c, d),$ if and only if $a d=b c .$ Then the number of ordered pairs which satisfy this equivalence relation with ordered pair $(4,3)$ is equal to :

Let $R\,= \{(x,y) : x,y \in N\, and\, x^2 -4xy +3y^2\, =0\}$, where $N$ is the set of all natural numbers. Then the relation $R$ is

In the set $A = \{1, 2, 3, 4, 5\}$, a relation $R$ is defined by $R = \{(x, y)| x, y$ $ \in  A$ and $x < y\}$. Then $R$ is

Consider the following two binary relations on the set $A= \{a, b, c\}$ : $R_1 = \{(c, a) (b, b) , (a, c), (c,c), (b, c), (a, a)\}$ and $R_2 = \{(a, b), (b, a), (c, c), (c,a), (a, a), (b, b), (a, c)\}.$ Then