Let f:A→B and g:B→C denote two functions. If the function
Let f:A→B and g:B→C denote two functions. If the function g∘f:A→C is a surjection and g is an injection then function f is,
- A.
Injection
- B.
Surjection
- C.
Bijection
- D.
None of these
Attempted by 78 students.
Show answer & explanation
Correct answer: B
Answer: f is a surjection.
Proof:
Take any element b in B. Then g(b) is an element of C.
Since g∘f is surjective, there exists a in A with g(f(a)) = g(b).
Because g is injective, g(f(a)) = g(b) implies f(a) = b.
Thus b is in the image of f. As this holds for every b in B, the image of f equals B, so f is surjective.
Remark: The original solution misstated that g is a surjection. That is not given. The correct argument uses the injectivity of g to lift preimages from g(b) back to b, proving surjectivity of f.