Math 220 Assignment 6
Due October 28th
Throughout, let A, B, C and D be sets, and let f : D → C be a function. In
particular, you no longer need to include these sentences in your proofs.
1. Let f : A → B and g : B → C be functions.
(a) Prove: if g ◦ f is surjective, then g is surjective.
(b) Prove: if g ◦ f is injective, then f is injective.
2. Let f : A → B and g : B → A be functions such that g ◦ f = idA . Prove
that g is surjective, and f is injective.
3. Let A ⊆ B. Prove that P (A) ⊆ P (B).
4. Prove that P (A) ∩ P (B) = P (A ∩ B).
5. Let T = {0, 1}, and let T A be the set of all functions f : A → T . Consider
the map F : T A → P (A) given by F (f ) = f −1 (0). Let G : P (A) → T A
be the map defined
as follows: for S ⊆ A, let GS : A → T be the
0 a∈S
map GS (a) =
Let G : P (A) → T A be the map defined
1 a 6∈ S
by G(S) = GS . Prove that F and G are inverses of each other. Recall:
f −1 (0) = {a ∈ A : f (a) = 0}.
(a) For every set S ⊆ A, prove that G−1
S (0) = S, an equality of sets.
Conclude that (F ◦ G) = idP (A) .
(b) For every function f : A → {0, 1}, prove that f = Gf −1 (0) , and
equality of functions. Conclude that (G ◦ F ) = idT A .
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