Class 12 Relations And Functions · Long Answer3 Markstypes_of_relations
Class 12 Maths · Relations And Functions · Long Answer #217
On A={1,…,7}, define aRb iff |a−b| is even. Prove equivalence.
✅ Step-by-Step Solution
Final Answer: An equivalence relation.
- Evenness of |a−b| is equivalent to a−b=2k for some integer k. The diagonal difference is 0, so R is reflexive.
- Changing a−b to b−a only changes k to −k, proving symmetry.
- If a−b=2k and b−c=2l, then a−c=2(k+l), proving transitivity.
- Hence R is an equivalence relation (its classes are the odd and even elements of A).
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Evenness of |a−b| is equivalent to a−b=2k for some integer k. The diagonal difference is 0, so R is reflexive.
- [1 mark] Changing a−b to b−a only changes k to −k, proving symmetry.
- [1 mark] If a−b=2k and b−c=2l, then a−c=2(k+l), proving transitivity.
- [0.5 marks] Hence R is an equivalence relation (its classes are the odd and even elements of A).
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