Final answer:
Statements about compositions of functions were examined: A true statement that if the composition is onto then f must be onto, and a true statement that if the composition is one-to-one then g must be one-to-one. Statement C about bijective composition requires additional context and is indeterminate.
Step-by-step explanation:
The question involves proving statements about functions and their compositions, specifically focusing on properties like being onto (surjective) and one-to-one (injective).
- If f∘g is onto, then f must also be onto. To show this, assume that f∘g is onto. For any element c in set C, there exists an element a in set A such that (f∘g)(a) = c. Since g(a) is in B, and f maps g(a) to c, every element in C is the image under f of some element in B, proving that f is onto.
- If f∘g is one-to-one, then g must also be one-to-one. To prove this, assume that g is not one-to-one. Then, there are elements a and a' in A with a ≠ a' such that g(a) = g(a'). Applying function f to both sides gives f(g(a)) = f(g(a')), which means f∘g(a) = f∘g(a'). However, this contradicts the assumption that f∘g is one-to-one. Hence, if f∘g is one-to-one, g must be one-to-one as well.
- For the last statement to prove If f∘g is a bijection, then g is onto if and only if f is one-to-one, the 'if and only if' portion makes this statement incomplete and context-dependent. Hence, this specific claim cannot be entirely proven as true or false in this abbreviated form and without additional context.
Based on the statements and proofs above, Statements A and B are true, while Statement C is indeterminate without further context.