Let R be a relation defined on as a R b if 2 a + 3 b is a multiple of 5, a , b . [2023]

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Solution

Q.
Let R be a relation defined on $ℕ$ as $a R b$ if $2 a + 3 b$ is a multiple of 5, $a, b \in ℕ$ . [2023]

1 symmetric but not transitive

2 not reflexive

3 an equivalence relation

4 transitive but not symmetric

Ans.
(3)

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

(i) $a R a \Rightarrow 5 a$ is a multiple of 5

R is a reflexive relation.

(ii) $a R b, 2 a + 3 b = 5 α$ (let)

Now, $b R a \Rightarrow 2 b + 3 a = 2 b + (\frac{5 α - 3 b}{2}) \cdot 3$

$= \frac{15}{2} α - \frac{5}{2} b = \frac{5}{2} (3 α - b) = \frac{5}{2} (2 a + 2 b - 2 α) = 5 (a + b - α)$

So, R is a symmetric relation.

(iii) $a R b \Rightarrow 2 a + 3 b = 5 α$ ; $b R c \Rightarrow 2 b + 3 c = 5 β$

Now, $2 a + 5 b + 3 c = 5 (α + β)$

$\Rightarrow 2 a + 5 b + 3 c = 5 (α + β) or 2 a + 3 c = 5 (α + β - b)$

$\Rightarrow a R c$

So, R is a transitive relation.

Hence Relation 'R' is an equivalence relation.

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