Class 12 Relations And Functions · Long Answer3 Markstypes_of_relations

Class 12 Maths · Relations And Functions · Long Answer #173

On A={0,1,…,12}, prove that aRb iff |a−b| is divisible by 3 is an equivalence relation.

✅ Step-by-Step Solution

Final Answer: An equivalence relation.

Divisibility of |a−b| by 3 is equivalent to a−b being an integer multiple of 3; every diagonal difference is 0.
Changing the order changes only the sign of a−b, so divisibility is preserved and R is symmetric.
If a−b=3k and b−c=3l, then a−c=3(k+l), proving transitivity.
Together with the zero diagonal differences, these checks prove R is an equivalence relation.

[0.5 marks] Divisibility of |a−b| by 3 is equivalent to a−b being an integer multiple of 3; every diagonal difference is 0.
[1 mark] Changing the order changes only the sign of a−b, so divisibility is preserved and R is symmetric.
[1 mark] If a−b=3k and b−c=3l, then a−c=3(k+l), proving transitivity.
[0.5 marks] Together with the zero diagonal differences, these checks prove R is an equivalence relation.

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