Question

Let $R$ be a relation defined on $N$ as a $Rb$ is $2a+3b$ 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)

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