HWI - ECE 3250 HOMEWORK ASSIGNMENT I Fall 2009 1. Find,...

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ECE 3250 HOMEWORK ASSIGNMENT I Fall 2009 1. Find, read, and understand a proof of the Schr¨ oder-Bernstein Theorem, which states: Let A and B be sets. If there exists an injective mapping f : A B and an injective mapping g : B A , then there exists a bijective mapping h : A B . If you look online, it might help to search for “Schroeder-Bernstein.” 2. Given a set A , first let P o ( A ) be the power set of A without the empty set — i.e., P o ( A ) is the set of all nonempty subsets of A . A mapping κ : P o ( A ) A is called a choice function if κ ( S ) S for all S ∈ P o ( A ). In other words, a choice function “picks out” an element of every nonempty subset of A . Show that a choice function cannot be an injective mapping if A has at least two elements. (Please don’t do this by referring to the relative cardinalities of the sets — it’s easier than that. What’s more interesting is the question: does a choice function exist for every set A ? The Axiom of Choice
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HWI - ECE 3250 HOMEWORK ASSIGNMENT I Fall 2009 1. Find,...

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