Sets, Relations and Function: Let R be the real line. Consider the following subsets of the plane… | AIEEE 2008

AIEEE 2008 (Offline)Question 1

Let $R$ be the real line. Consider the following subsets of the plane $R\times R$ : $S=\{(x,y):y=x+1\text{ and }0<x<2\}$ $T=\{(x,y):x-y\text{ is an integer}\}$ Which one of the following is true ?

Sets, Relations and Function

Neither $S$ nor $T$ is an equivalence relation on $R$Both $S$ and $T$ are equivalence relation on $R$$S$ is an equivalence relation on $R$ but $T$ is not$T$ is an equivalence relation on $R$ but $S$ is not✓ Correct answer

Solution

$S$ is not reflexive since $(x,x)\notin S$ ($y=x+1\ne x$). $T$: $x-x=0\in\mathbb Z$, $x-y\in\mathbb Z\Rightarrow y-x\in\mathbb Z$, and sums of integers are integers, so $T$ is an equivalence relation.

Sets, Relations and Function - JEE Mathematics question (AIEEE 2008 (Offline))

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