Practice Problems on Sets Key

# Practice Problems on Sets Key - elements These are all...

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Practice Problems on Sets – Key Textbook: Mathematical Structures for Computer Science: A Modern Treatment of Discrete Structures   (5     th     Edition     ) J. Gersting, W.H. Freeman and Company: New York , NY . 2003. Page 179, #9 a) 2 b) 2 c) 1 d) 3 e) 3 Page 179, #10 a) T b) T c) F d) T e) T f) F g) F h) T Page 180, #20 a) Base: n = 2. A set with 2 elements has exactly 1 subset with 2 elements. 2(2 - 1) / 2 = 1 (TRUE) Assume: Any set with k elements has k(k-1)/2 subsets with exactly 2 elements WTS: Any set with k+1 elements has (k+1)k/2 subsets with exactly 2 elements. Let S be set with k+1 elements Let x S Remove x from S giving a set of k elements By inductive hypothesis, S - {x} has k(k-1)/2 subsets with exactly 2

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Unformatted text preview: elements. These are all 2-element subsets of S that do not include x All 2-element subsets of S can be found by pairing x in turn with each of the remaining k elements, giving k subsets. (QED) Page 184, #49 a) B ⊆ A b) A ⊆ B c) A = {} d) B ⊆ A e) A = B f) A = B Page 185, #63 a) A ∪ A = {x | x ∈ A or x ∈ A} = { x | x ∈ A} = A b) A ∩ A = {x | x ∈ A and x ∈ A} = {x | x ∈ A} = A c) A ∩ ∅ = {x | x ∈ A and x ∈ {} but since no x is a member of {}, this is the empty set. d) A ∪ S ⊆ S because A ⊆ S; S ⊆ A ∪ S by Exercise 52. e) x ∈ (A′)′ <--> x not in A′ <--> x not in {y | y not in A} <--> x ∈ A...
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Practice Problems on Sets Key - elements These are all...

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