Class 12 Relations And Functions · Very Long Answer5 Markstypes_of_relations
Class 12 Maths · Relations And Functions · Very Long Answer #186
On R, let xRy iff xy is irrational. Test the three relation properties.
✅ Step-by-Step Solution
Final Answer: Symmetric, but neither reflexive nor transitive.
- The definition tests irrationality of the product, including when the two inputs coincide.
- At x=0, x²=0 is rational, so xRx fails; reflexivity does not hold.
- If xy is irrational, yx is the same number and is irrational; hence R is symmetric.
- 1R√2 and √2R3 hold since √2 and 3√2 are irrational, but 1R3 fails since 3 is rational. Thus R is not transitive.
- Therefore R is symmetric only among these properties, and is not an equivalence relation.
📋 CBSE Stepwise Marking Scheme
- [1 mark] The definition tests irrationality of the product, including when the two inputs coincide.
- [1 mark] At x=0, x²=0 is rational, so xRx fails; reflexivity does not hold.
- [1 mark] If xy is irrational, yx is the same number and is irrational; hence R is symmetric.
- [1 mark] 1R√2 and √2R3 hold since √2 and 3√2 are irrational, but 1R3 fails since 3 is rational. Thus R is not transitive.
- [1 mark] Therefore R is symmetric only among these properties, and is not an equivalence relation.
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