Let [A=2,3,6,8,9,11] and [B=1,4,5,10,15].

Question

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

Accepted Answer

$$(a, b) R(c, d) \Rightarrow 3 a d-7 b c \in$$ even

For reflexive

$$(a, b) R(a, b) \Rightarrow 3 a b-7 b a=-4 a b \in$$ even

For symmetric

$(a, b) R(c, d)$

then

$(c, d) R(a, b)=3 b c-7 a d$

$$\in$$ even

Now check for transitive

$$\begin{aligned}
& \[\begin{gathered} (a, b) R(c, d) \text { and }(c, d) R(e, f) \text { then }(a, b) R(e, f) \ 3 a d-7 b c=2 m \quad \Rightarrow (2,5) R(6,8) \text { and } \ 3 c f-7 e d=2 n \quad (6,8) R(9,4) \ \text { then } 3 a f-7 e b \neq \text { even } \notin(2,5) R(9,4) \ \Rightarrow \text { Not transitive option }(3) \end{gathered}\end{aligned}$$
\]

Solutions · 1