Class 12 Relations And Functions · Very Long Answer5 Markstypes_of_relations
Class 12 Maths · Relations And Functions · Very Long Answer #227
On A={0,1,…,12}, define aRb iff 4 divides a−b. Prove equivalence and find [1].
✅ Step-by-Step Solution
Final Answer: Equivalence; [1]={1,5,9}.
- For each a∈A, a−a=0 is divisible by 4, establishing reflexivity.
- If a−b=4k, then b−a=4(−k), proving symmetry.
- If a−b=4k and b−c=4l, then a−c=4(k+l), proving transitivity and hence equivalence.
- The class of 1 consists of x∈A with x−1=4k. Within 0≤x≤12 this gives k=0,1,2 and x=1,5,9.
- Therefore R is an equivalence relation and [1]={1,5,9}.
📋 CBSE Stepwise Marking Scheme
- [1 mark] For each a∈A, a−a=0 is divisible by 4, establishing reflexivity.
- [1 mark] If a−b=4k, then b−a=4(−k), proving symmetry.
- [1 mark] If a−b=4k and b−c=4l, then a−c=4(k+l), proving transitivity and hence equivalence.
- [1 mark] The class of 1 consists of x∈A with x−1=4k. Within 0≤x≤12 this gives k=0,1,2 and x=1,5,9.
- [1 mark] Therefore R is an equivalence relation and [1]={1,5,9}.
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