Class 12 Relations And Functions · Very Long Answer5 Markstypes_of_relations
Class 12 Maths · Relations And Functions · Very Long Answer #185
On N×N, define (a,b)R(c,d) iff ad=bc. Prove equivalence.
✅ Step-by-Step Solution
Final Answer: An equivalence relation.
- Positive denominators allow ad=bc to be rewritten as a/b=c/d.
- Each ratio equals itself; equivalently ab=ba, so reflexivity holds.
- If a/b=c/d, then c/d=a/b, proving symmetry.
- If a/b=c/d and c/d=e/f, then a/b=e/f and af=be, proving transitivity.
- The relation is reflexive, symmetric and transitive, hence an equivalence relation.
📋 CBSE Stepwise Marking Scheme
- [1 mark] Positive denominators allow ad=bc to be rewritten as a/b=c/d.
- [1 mark] Each ratio equals itself; equivalently ab=ba, so reflexivity holds.
- [1 mark] If a/b=c/d, then c/d=a/b, proving symmetry.
- [1 mark] If a/b=c/d and c/d=e/f, then a/b=e/f and af=be, proving transitivity.
- [1 mark] The relation is reflexive, symmetric and transitive, hence an equivalence relation.
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