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RELATIONS AND FUNCTIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsRELATIONS AND FUNCTIONS3 Marks
Question
Prove that the relation R in the set $\text{A } = (1, 2, 3, 4, 5)$ given by $\text{R} = (\text{a, b)} : |\text{a-b|} \text{is even},$ is an equivalence relation.
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Answer

$\text{(i) for all a} \in \text{A, (a ,a)} \in \text{R} \therefore | \text{a - a}| = \text{o is even} $$\therefore \text{R is reflexive in A}$
$\text{(ii) for all a, b} \in \text{A, (a, b)} \in \text{R} \Rightarrow \text{b, a)} \in \text{R} \because \text{if | a -b| is even then|b- a| is also even } \Rightarrow \text{R is symmetric in A} $
$\text{(iii) for all a, b, c} \in \text{A}$
$\text{(a, b)} \in \text{R and (b,c)} \in \text{R then (a, c) } \in \text{R}$
$\because \text{|a - b| is even, | b - c| is even, then |a - c| will also be even}$
$\text{Hence, R is an equivalence relation in A}$

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