Then s t 1 2 3 4 and s 3 t 1 and s t

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Unformatted text preview: ot;may" we only need an example. One example is the set of integers Z, which is a subset of the real numbers. Comment: In fact, any infinite set S contains a countable subset. Just pick one element as x1. Then pick x2 ∈ S – {x1}, x3 ∈ S – {x1, x2}, etc. This process gives an infinite list of elements of S, since one runs out of elements only if S is finite. Then the set of xi is countable. e) If S is a finite set, then S contains the same number of elements as the collection of subsets of S. FALSE. For counterexample, see Problem 4.1 Corrected statement #1: If S is a finite set, S contains fewer elements than P(S), the set of subsets of S. Proof: Let #S = n, and let S = {s1, … , sn}. Then {{}, {s1}, …, {sn}} is a subset of P(S) that has n+1 elements. So #S = n < n+1 ≤ #P(S). Corrected statement #2: If S is a finite set with n elements, then S has 2n subsets. Proof. This is a much stronger and better statement. We have seen this in examples and will prove it soon as an example of mathematical induction. If some students stated and proved this statement using induction (formally or informally), that is very good. Problem 4.5: Gemignani, Section 6.5 # 2 Compute each of the following sums of cardinal numbers by means of the procedure described in Definition 6.4. Note: I am not sure how to create an over ­bar with Word, so I am using underline instead. (Note: (a) (b) and (c) are set up in 3 different ways just to make it more interesting. Any of these methods wi...
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