Let $A=\{2,3,6,8,9,11\}$ and $B=\{1,4,5,10,15\}$. Let $R$ be a relation on $A \times B$ defined by $(a, b)…
Let $A=\{2,3,6,8,9,11\}$ and $B=\{1,4,5,10,15\}$. Let $R$ be a relation on $A \times B$ defined by $(a, b) R(c, d)$ if and only if $3 a d-7 b c$ is an even integer. Then the relation $R$ is
an equivalence relation.
reflexive and symmetric but not transitive.
transitive but not symmetric.
reflexive but not symmetric.
Solution
$\begin{array}{lc}\mathrm{A}=\{2,3,6,8,9,11\} & (\mathrm{a}, \mathrm{b}) \mathrm{R}(\mathrm{c}, \mathrm{d}) \\ \mathrm{B}=\{1,4,5,10,15\} & 3 \mathrm{ad}-7 \mathrm{bc}\end{array}$
Reflexive : $(\mathrm{a}, \mathrm{b}) \mathrm{R}(\mathrm{a}, \mathrm{b})$
$\Rightarrow 3 \mathrm{ab}-7 \mathrm{ba}=-4 \mathrm{ab}$ always even so it is reflexive.
Symmetric : If $3 \mathrm{ad}-7 \mathrm{bc}=$ Even
Case-I : odd odd
Case-II : even even
(c, d) $\mathrm{R}(\mathrm{a}, \mathrm{b}) \Rightarrow 3 \mathrm{bc}-3 \mathrm{ab}$
Case-I : odd odd
Case-II : even even
so symmetric relation
Transitive :
Set $(3,4) R(6,4)$ Satisfy relation
Set $(6,4) R(3,1)$ Satisfy relation
but $(3,4) \mathrm{R}(3,1)$ does not satisfy relation so not transitive.