hw3cs - CSE 20 Discrete Mathematics Fall 2010 Problem Set 3...

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Unformatted text preview: CSE 20: Discrete Mathematics Fall 2010 Problem Set 3 Instructor: Daniele Micciancio Due on: Wed. Oct. 20, 2010 Problem 1 (10 points) Recall the well-ordering principle presented in class on Monday: Fact 1. (Well Ordering principle) Every non-empty set S of positive integers contains a smallest element. In symbols: “ ( S ⊆ Z + ) ∧ ( S 6 = ∅ ) = ⇒ ∃ x ∈ S. ∀ y ∈ S.x ≤ y ”. (a) Use the well-ordering principle to prove that every positive integer number is either even or odd. [Hint: use the definition of “even(n)= ∃ x ∈ Z .n = 2 x ” and “odd(n)= ∃ x ∈ Z .n = 2 x + 1 ”, and consider the set S = { n ∈ Z + | ¬ even ( n ) ∧ ¬ odd ( n ) } of all positive integers that are neither even nor odd.] (b) Use part (a) to prove that every integer is either even or odd. (The difference with part (a) is that this time we consider also negative numbers.) Note: You can work on part (b) independently from part (a), i.e., even if you don’t solve (a), you can assumeNote: You can work on part (b) independently from part (a), i....
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This note was uploaded on 01/08/2011 for the course CSE cse105 taught by Professor Cs during the Fall '10 term at UCSD.

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hw3cs - CSE 20 Discrete Mathematics Fall 2010 Problem Set 3...

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