Sets, Relations & Functions
General
Grade 11
Question:
<p>Which are correct?<br>(A) One-one self-map is onto (B) Onto self-map is one-one (C) g∘f injective ⟹ f injective (D) |A|=3,|B|=2: #functions = 8</p>
If f is one-one from A to A, then f is onto.
If f is onto from A to A, then f is one-one.
If g∘f is injective, then f must be injective.
If |A|=3 and |B|=2, number of functions A o B is 8.
Step-by-Step Solution
Key Concept: General
<div class="solution"><p>(A) False for infinite sets (e.g. n↦n+1 on ℕ)</p><p>(B) False for infinite sets</p><p>(C) If f(x₁)=f(x₂), then (g∘f)(x₁)=(g∘f)(x₂); injectivity of g∘f forces x₁=x₂. ✓</p><p>(D) Each of 3 elements has 2 choices: 2³=8 ✓</p><p><strong>Answer: (C),(D)</strong></p><div class="trap-box"><strong>Trap:</strong> (A) and (B) are true only for finite sets. For infinite sets both fail.</div><div class="key-concept"><strong>Key Concept:</strong> g∘f injective ⟹ f injective (standard composition rule)</div></div>
Correct Answer: C,D