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RELATIONS AND FUNCTIONS — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSRELATIONS AND FUNCTIONS3 Marks
Question
Let n be a fixed positive integer. Define a relation R in Z as follows $\forall\ \text{a},\ \text{b}\in\text{Z},$ aRb if and only if a - b is divisible by n. Show that R is an equivalance relation.
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Answer

Given that, $\forall\ \text{a},\ \text{b}\in\text{Z},$ aRb if and only if a - b is divisible by n. Now, I. Reflexive aRa ⇒ (a - a) is divisible by n, which is true for any integer 'a' as ‘0’ is divisible by n.Hence, R is reflexive.
II. Symmetric.
aRb ⇒ a - b is divisible by n. ⇒ -(b - a) is divisible by n. ⇒ (b - a) is divisible by n. ⇒ bRaHence, R is symmetric.
III. Transitive.
Let aRb and bRc
⇒ (a - b) is divisible by n and (b - c) is divisible by n. ⇒ (a - b) + (b - c) is divisible by n. ⇒ (a - c) is divisible by n. ⇒ aRcHence, R is transitive.
So, R is an equivalence relation.

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