Which of the following is an equivalence relation-1 xy3 x-y is divisible by 54 x divides y

x lt; y ⇒ y gt; x(false) ⇒ not symmetric x gt; y ⇒ y lt; x(false) ⇒ not symmetric x y divisible by 5 ⇒ reflexive x x=0 divi ...

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Standard XII

Mathematics

Equivalence Relation

Which of the ...

Question

Which of the following is an equivalence relation?
1) $x < y$
2) $x > y$
3) $x - y$ is divisible by $5$
4) $x$ divides $y$

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Solution

$x < y \Rightarrow y > x (f a l s e) \Rightarrow$ not symmetric
$x > y \Rightarrow y < x (f a l s e) \Rightarrow$ not symmetric
$x - y$ divisible by $5 \Rightarrow$ reflexive
$x - x = 0$ divisible by $5 \Rightarrow$ Symmetric
$x - y$ is divisible by $5$ and $y - z$ is divisible by $5$
so, $x - z$ is also divisible by $5$
$∴ e q u i v a l e n t$

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Which of the following is an equivalence relation?1)x<y2)x>y3)x−y is divisible by 54)x divides y

x<y⇒y>x(false)⇒not symmetricx>y⇒y<x(false)⇒not symmetricx−y divisible by 5⇒ reflexivex−x=0 divisible by 5⇒ Symmetricx−y is divisible by 5 and y−z is divisible by 5so, x−z is also divisible by 5∴equivalent

Which of the following is an equivalence relation?
1) $x < y$
2) $x > y$
3) $x - y$ is divisible by $5$
4) $x$ divides $y$