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5. In this problem we shall prove the inductive step in the proof by math- ematical induction of the formula in (1). Let P(n) be the statement that (1) is true for any finite sets A1- A2,..., An. Suppose P(4) true. Let A1 1.Az.An,As.As be finite sets. Then (Un/-((u)/und DAING Explain why each of the above equalities of sets is true. Prove that the set on the last line is equal to 04JCNs Hint: The nonempty subsets of N° are exactly of two types: (b) J =10(5} where CNA In this case, we have two subtypes: i. J= IU {5}, where 0# ICNA, ii. J={5}. 6. Let f Y be an injection. Let IO E X. Define g X - (Io} - y {f(=o)} by g(x) /(x) for zex-(20). 2 (a) Prove that is well defined. That is, for any x € X (Io) g(x) {f(=o)} (b) Prove that g is an injection. 7. Let P(n) be the statement: Ifm E2+ and / N.. N.. is an injection, then m n. Suppose E that P(k) is true. Let me z+ and let / N, + NA+1 be an injection. Define by g(i) f(i) for if Nm-1: (a) Prove that a is an injection (b) Using the Addition Principle, prove that |Nx+1 {((m)}| k: (c) Using g and part (b), show that there exists an injection h : (d) Prove that P(k 1) is true. 8. Let f x Y be surjection. Let ID E X. Define h x {=o} + Y by h(x) f(x) for X (Io} Suppose that h is not surjection. Define g X-{x > {((xx)} by 9(2)=h(7)=f(f) x {xo} (a) Prove that g is well defined. That is, for any x E x - (Io) (b) Prove that g is surjection. 9. Let P(n) be the statement: If m EZ+ -> N.. is asurjection then m Sn. Suppose AEZ+ is such that P(k) is true. Let me z+ and let f: Ne+1 Nm be surjection. Define by h(i) /(i) for Ng (a) Prove that if is surjection, then 772 3 (b) Suppose h is not a surjection. Prove that g Nr N, - {/(k+1)} by g(i)=h(i) =f(i)forieNa is a well defined surjection. (c) Assuming the same hypothesis as (b), prove that there exists a surjection j (d) Prove that P(A 1) is true.

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