"Let R be a relation defined on $\mathbb{N}$ as $a\mathrm{R}b$ if $2a+3b$ is a multiple of $5,a,b\in \mathbb{N}$. Then R is"

Question

Accepted Answer

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a R b if 2a + 3b = 5m, m $\in$ $l$

(1) $(a,a) \in R$ as $2a + 3a = 5a,a \in N$

Hence, R is reflexive

(2) If $(a,b) \in R$ then $2a + 3 = 5m$

Now, $5(a + b) = 5n$

$3a + 2b + 2a + 3b = 5n$

$\therefore$ $3a + 2b = 5(n - m)$

$\therefore$ $(b,a) \in R$

$\therefore$ R is symmetric

(3) If $(a,b) \in R$ and $(b,c) \in R$ then

$2a + 3b = 5m,2b + 3c = 5n$

$\Rightarrow 2a + 5b + 3c = 5(m + n)$

$\Rightarrow 2a + 3c = 5(m = n - b)$

$\therefore$ $(a,c) \in R$

$\therefore$ R is transitive

Hence, R is equivalence relation.

Option (1) is correct.

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