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Exercise III Consider sets A, B, C, and functions $f: A \to B$ and $g: B \to C$. 1. Show that if $g \circ f$ is 1-1, then $f$ is 1-1. 2. Show that if $g \circ f$ is onto, then $g$ is onto. 3. Consider two functions $u: A \to C$ and $v: C \to A$. Assume that $v \circ u = id_A$ and $u \circ v = id_C$. Show that $u$ is a bijection and that $v = u^{-1}$. Use Questions 1 and 2. 4. Assume that $f$ and $g$ are bijective. Show that $g \circ f$ is bijective and that $(g \circ f)^{-1} = f^{-1} \circ g^{-1}$. Use Question 3.

          Exercise III
Consider sets A, B, C, and functions $f: A \to B$ and $g: B \to C$.
1. Show that if $g \circ f$ is 1-1, then $f$ is 1-1.
2. Show that if $g \circ f$ is onto, then $g$ is onto.
3. Consider two functions $u: A \to C$ and $v: C \to A$. Assume that $v \circ u = id_A$ and $u \circ v = id_C$. Show
that $u$ is a bijection and that $v = u^{-1}$. Use Questions 1 and 2.
4. Assume that $f$ and $g$ are bijective. Show that $g \circ f$ is bijective and that $(g \circ f)^{-1} = f^{-1} \circ g^{-1}$.
Use Question 3.

Exercise III
Consider sets A, B, C, and functions f: A → B and g: B → C.
1. Show that if g ∘ f is 1-1, then f is 1-1.
2. Show that if g ∘ f is onto, then g is onto.
3. Consider two functions u: A → C and v: C → A. Assume that v ∘ u = idA and u ∘ v = idC. Show
that u is a bijection and that v = u^-1. Use Questions 1 and 2.
4. Assume that f and g are bijective. Show that g ∘ f is bijective and that (g ∘ f)^-1 = f^-1∘ g^-1.
Use Question 3.

Added by Kathleen C.

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