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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 . 1