Question

Let $R$ be a relation defined on $N$ as a $R\, b$ is $2 a+3 b$ is a multiple of $5, a, b \in N$. Then $R$ is

• A. not reflexive
• B. transitive but not symmetric
• C. symmetric but not transitive
• D. an equivalence relation

Answer 1

Answer: (D)

$a R a \Rightarrow 5 a$ is multiple it 5
So reflexive
$a R b \Rightarrow 2 a +3 b =5 \alpha \text {, }$
Now $b \,R \,a$
$2 b+3 a =2 b+\left(\frac{5 \alpha-3 b}{2}\right) \cdot 3 $
$=\frac{15}{2} \alpha-\frac{5}{2} b=\frac{5}{2}(3 \alpha-b)$
$ =\frac{5}{2}(2 a+2 b-2 \alpha)$
$=5(a+b-\alpha)$
Hence symmetric
$ \text { a R b } \Rightarrow 2 a+3 b=5 \alpha$
$ \text { b R c } \Rightarrow 2 b+3 c=5 \beta $
$ \text { Now } 2 a+5 b+3 c=5(\alpha+\beta)$
$ \Rightarrow 2 a +5 b +3 c =5(\alpha+\beta) $
$ \Rightarrow 2 a+3 c=5(\alpha+\beta-b) $
$ \Rightarrow a R c $
Hence relation is equivalence relation.