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

Option: 1

\mathrm{\alpha=14}


Option: 2

\mathrm{\alpha \text { is a multiple of } 4}


Option: 3

\mathrm{4 \text { is the remainder when } \alpha \text { is divided by } 10}


Option: 4

\mathrm{4 \text { is the remainder when } \alpha \text { is divided by } 7}


Answers (1)

best_answer

\begin{aligned} & \text{for R to be reflexive} \mathrm{ \Rightarrow x R x} \\ & \mathrm{\Rightarrow 3 x+\alpha x=7 x \Rightarrow(3+\alpha) x=7 k} \\ & \mathrm{\Rightarrow 3+\alpha=7 \lambda \Rightarrow \alpha=7 \lambda-3=7 N+4,}\\ & \mathrm{k, \lambda, N \in I}\\ & \therefore \text{ when} \alpha \text{divided by 7 , remainder is 4.}\\ & \text{R to be symmetric} \mathrm{x R y \Rightarrow y R x}\\ & \mathrm{3 x+\alpha y=7 N_{i}, 3 y+\alpha x=7 N_{2}}\\ & \mathrm{\Rightarrow(3+\alpha)(x+y)=7\left(N_{1}+N_{2}\right)=7 N_{3} }\\ & \text{which holds. when} 3+\alpha \text{ is multiole of 7}\\ & \therefore \alpha=7 N+4 (\text{as did earlier})\\ &\text{R to be transitive}\\ & \mathrm{x R y \; \& \; y R z \Rightarrow x R z.} \end{aligned}

\begin{aligned} & \mathrm{3 x+2 y=7 N_{1} \text { \&f } 3 y+2 z=7 N_{2} \text { and }} \\ & \mathrm{3 x+\alpha z=7 N_{3}} \\ & \mathrm{\therefore 3 x+7 N_{2}-3 y=7 N_{3} }\\ & \mathrm{\therefore 7 N_{1}-\alpha y+7 N_{2}-3 y=7 N_{3}} \\ & \mathrm{\therefore 7\left(N_{1}+N_{2}\right)-(3+\alpha) y=7 N_{3}} \\ & \mathrm{\therefore(3+\alpha) y=7 N} \end{aligned}

\begin{aligned} & \text{which is true again when}\; 3+\alpha \; \text{divisib'e by. 7} ; \\ & \text{i.e. when}\; \alpha \; \text{divided by 7 , remainder is 4.} \end{aligned}

Answer is D

Posted by

Ritika Kankaria

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