Sets, Relations and Function: For alphainmathbfN , consider a relation R on mathbfN given by… | JEE Main 2022

JEE Main 2022 (Online) 28th July Morning ShiftQuestion 1

For $\alpha\in\mathbf{N}$, consider a relation $R$ on $\mathbf{N}$ given by $R=\{(x,y):3x+\alpha y\text{ is a multiple of }7\}$. The relation $R$ is an equivalence relation if and only if :

Sets, Relations and Function

$\alpha=14$$\alpha$ is a multiple of $4$$4$ is the remainder when $\alpha$ is divided by $10$$4$ is the remainder when $\alpha$ is divided by $7$✓ Correct answer

Solution

Reflexive: $(3+\alpha)x\equiv0\pmod7$ for all $x\Rightarrow\alpha\equiv4\pmod7$. Conversely if $\alpha\equiv4$: $3x+4y\equiv0\Rightarrow3x\equiv-4y\Rightarrow x\equiv-20y\equiv y\pmod7$, so $R$ is $x\equiv y\pmod 7$, an equivalence relation. So iff $\alpha$ leaves remainder $4$ on division by $7$: option D.

Sets, Relations and Function - JEE Mathematics question (JEE Main 2022 (Online) 28th July Morning Shift)

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