Let f : A -> B be an onto (or surjective) function, where A and B are nonempty sets. Define an equivalence relation ~ on the set A as: a_1 ~ a_2 if f(a_1) = f(a_2) where a_1, a_2 in A. Let E = {[a] | a in A} be the set of all the equivalence classes under ~. Define a new mapping F : E -> B as F([a]) = f(a) for all equivalence classes [a] in E. Which of the following statements is/are TRUE? A. F is NOT well-defined. B. F is an onto (or surjective) function. C. F is an injective function. D. F is a bijective function.

GATE 2023 · Discrete Mathematics · Functions · medium

Answer: B, C, D

1. Verify F is well-defined: If [a_1] = [a_2], then a_1 ~ a_2, so f(a_1)=f(a_2) by definition of ~. Hence F([a_1]) = F([a_2]). F is well-defined. Statement A is FALSE.
2. Verify F is injective: F([a_1]) = F([a_2]) means f(a_1) = f(a_2) means a_1 ~ a_2 means [a_1] = [a_2]. So F is injective. Statement C is TRUE.
3. Verify F is surjective and conclude: For any b in B, since f is surjective there exists a in A with f(a)=b. Then F([a])=f(a)=b. So F is surjective (Statement B is TRUE). Since F is both injective and surjective, F is bijective (Statement D is TRUE).