Class 12 Relations And Functions · Very Long Answer5 Markstypes_of_relations
Class 12 Maths · Relations And Functions · Very Long Answer #181
On R, define aRb iff a−b+√3 is irrational. Test reflexivity, symmetry and transitivity.
✅ Step-by-Step Solution
Final Answer: Reflexive, but neither symmetric nor transitive.
- Use the defining irrationality test separately on diagonal, reversed and composed pairs.
- For every a, a−a+√3=√3 is irrational, so R is reflexive.
- Take a=√3,b=0. The forward value is 2√3, irrational, while the reverse value is 0, rational; symmetry fails.
- Take a=0,b=√3/2,c=√3. The first two differences plus √3 equal √3/2, irrational; the composed value a−c+√3=0 is rational. Thus transitivity fails.
- Therefore only reflexivity holds; R is not an equivalence relation.
📋 CBSE Stepwise Marking Scheme
- [1 mark] Use the defining irrationality test separately on diagonal, reversed and composed pairs.
- [1 mark] For every a, a−a+√3=√3 is irrational, so R is reflexive.
- [1 mark] Take a=√3,b=0. The forward value is 2√3, irrational, while the reverse value is 0, rational; symmetry fails.
- [1 mark] Take a=0,b=√3/2,c=√3. The first two differences plus √3 equal √3/2, irrational; the composed value a−c+√3=0 is rational. Thus transitivity fails.
- [1 mark] Therefore only reflexivity holds; R is not an equivalence relation.
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