Class 12 Relations And Functions · Long Answer3 Markstypes_of_relations
Class 12 Maths · Relations And Functions · Long Answer #172
Prove that aRb iff 2 divides a−b is an equivalence relation on Z.
✅ Step-by-Step Solution
Final Answer: An equivalence relation.
- For any integer a, a−a=0 is divisible by 2, so R is reflexive.
- If a−b=2k for an integer k, then b−a=2(−k); hence R is symmetric.
- If a−b=2k and b−c=2l, then a−c=2(k+l); hence R is transitive.
- R is reflexive, symmetric and transitive, so it is an equivalence relation.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] For any integer a, a−a=0 is divisible by 2, so R is reflexive.
- [1 mark] If a−b=2k for an integer k, then b−a=2(−k); hence R is symmetric.
- [1 mark] If a−b=2k and b−c=2l, then a−c=2(k+l); hence R is transitive.
- [0.5 marks] R is reflexive, symmetric and transitive, so it is an equivalence relation.
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