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Unformatted text preview: min(m, n) ≥ #(S ∩ T). Then the same example as in (a) shows that in some cases, min(m, n) = #(S ∩ T), so this is the largest number.. d) What is the smallest number of elements that S ∩ T can contain? Answer: If S and T are disjoint, then S ∩ T is the empty set and has 0 elements. e) Find a formula for the number of elements of S ∪ T in terms of the number of elements of S, T and S ∩ T. Answer: In class on Wednesday 1/25, we proved this formula: #(S ∪ T) = #S + #T – #(S ∩ T) Notice that most of (a) – (d) follows from this formula. Problem 4.4: Gemignani, Section 6.3 # 3 Determine whether each of the statements is true or false. If a statement is true, supply a proof. If a statement I false, correct the statement and prove the corrected statement. a) Any two subsets of the set of positive even integers contain the same number of elements. This is clearly FALSE, since any two finite subsets with different cardinal numbers (for example {2} and {2, 4}) do not contain the same number of elements. One corrected version that is true: Any two infinite subsets of the set of positive even integers contain the same number of elements. Proof. The proof for Proposition 3 will also prove this statement. b) Any two infinite subsets of the set of integers contain the same number of elements. This is TRUE. Proof: We will prove that any infinite subset of the set of integers is countable. Let S be such a subset and let S1 be the set of elements in S that are ≥ 0 and let S2 be the set of elements of S that are < 0. Then S = S1∪S2 . By the reasoning of the proof of Proposition 3, we list the elements of S1 in their natural order and this pairs up the...
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 Winter '11
 JamesKing
 Math, Sets

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