CBSE Class 12 Mathematics 2026 Supplementary Paper 65(B)/7 · Q325 Marksequivalence_relations
CBSE Class 12 Mathematics 2026 Supplementary Paper 65(B)/7 · Question 32 (Relations And Functions)
On ordered pairs of positive natural numbers define (a,b)R(c,d) when ad=bc. Prove R is an equivalence relation. OR Prove f:ℝ∖{2}→ℝ∖{1}, f(x)=(x−1)/(x−2), is one-one and onto.
Alternative 1 5 Marks
On ordered pairs of positive natural numbers define (a,b)R(c,d) when ad=bc. Prove R is an equivalence relation.
Answer: R is an equivalence relation.
- Because b and d are positive, ad=bc is equivalent to a/b=c/d.
- Equality of ratios is reflexive and symmetric. If a/b=c/d and c/d=e/f then a/b=e/f, giving transitivity.
- Therefore R has all three required properties.
Alternative 2 5 Marks
Prove f:ℝ∖{2}→ℝ∖{1}, f(x)=(x−1)/(x−2), is one-one and onto.
Answer: f is bijective.
- For an arbitrary target y≠1 solve y=(x−1)/(x−2).
- The unique candidate is x=(2y−1)/(y−1), which is never 2. Substitution returns y.
- Every target has exactly one preimage, proving both properties.