Let's solve the problem step by step. We are given three functions $f: A \to B$, $g: B \to C$, and $h: C \to D$. We need to prove three statements based on the composition of these functions.

1. Prove: If $f$ and $g$ are one-to-one, then $g \circ f$ is one-to-one.

Proof:

• Let $f$ and $g$ be injective (one-to-one functions).
• To prove that $g \circ f$ is injective, assume $g(f(x_1)) = g(f(x_2))$ for some $x_1, x_2 \in A$.
• Since $g$ is injective, we can conclude that $f(x_1) = f(x_2)$.
• Since $f$ is injective, $x_1 = x_2$.
• Thus, $g \circ f$ is injective (one-to-one).

2. Prove: If $f$ and $g$ are onto, then $g \circ f$ is onto.

Proof:

• Let $f$ and $g$ be surjective (onto functions).
• To prove that $g \circ f$ is surjective, take any $c \in C$. Since $g$ is surjective, there exists some $b \in B$ such that $g(b) = c$.
• Since $f$ is surjective, there exists some $a \in A$ such that $f(a) = b$.
• Therefore, $g(f(a)) = g(b) = c$, which shows that for every $c \in C$, there exists an $a \in A$ such that $g(f(a)) = c$.
• Thus, $g \circ f$ is surjective (onto).

3. Prove: $(h \circ g) \circ f = h \circ (g \circ f)$.

Proof:

• We need to show that $(h \circ g) \circ f = h \circ (g \circ f)$.
• By the definition of composition of functions, we have:
  • $(h \circ g) \circ f (x) = h(g(f(x)))$,
  • $h \circ (g \circ f)(x) = h(g(f(x)))$.
• Since both sides simplify to $h(g(f(x)))$, the equality holds.
• Therefore, $(h \circ g) \circ f = h \circ (g \circ f)$.

These are the required proofs for the given statements.

Do you want more details on any of the steps or have any questions?

Here are some related questions for further practice:

1. Can you prove that the composition of two surjective functions is surjective?
2. How would you define the inverse of a composition of two injective functions?
3. What is the general property of function composition with respect to associativity?
4. If $g$ is not surjective but $f$ is, is $g \circ f$ surjective?
5. Can a function be injective but not surjective? Provide an example.

Tip: Always pay attention to how function properties (injectivity and surjectivity) transfer through composition, as these are fundamental in proving larger properties of composed functions.