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20_proofs

# 20_proofs - Introduction to Computers and Programming Prof...

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Introduction to Computers and Programming Prof. I. K. Lundqvist Lecture 20 2 Proof by Truth Table x Æ y and ( ¬ x) y are logically equivalent 1 0 1 1 1 0 0 0 0 1 1 1 1 1 0 1 1 1 0 0 ( ¬ x) y ¬ x x Æ y y x May 5 2004 •Propos

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3 Definitions Even An integer n is even, iff n =2 k for some integer k . n is even ↔∃ k such that n = 2 k Odd An integer n is odd, iff n =2 k +1 for some integer k . n is odd ↔∃ an integer k such that n = 2 k +1 Divisible An integer n is divisible by m , iff n and m are integers such that that there is an integer k such that mk = n n is divisible by m ↔∃ an integer k such that mk = n 4 The sum of two even integers is even the conclusion (end) unwind definitions an integer 1. Rewrite as a condition (using if, then) 2. Write the hypothesis (beginning) and 1. For beginning: establish notation and 2. For end: unwind definitions backwards 3.…and…wa
5 The sum of two even integers is even 1. [conditional] Proof: 2. Let x and y be integers [hypothesis] [conclusion] 5. y is even, so 2|y 6. There is an integer, e.g., b, with y=2b [as above] 7. 4. There is an integer, e.g., a, with x=2a [def. of divisible] 8. 3. x is even, so 2|x 9. (x+y) is even, so 2|(x+y) 6 Direct Proof simple combination of existing theorems restated the theorem and beginning of the proof your argument. If x and y are even integers, then x+y is even 10. x + y is even [likewise for y] (x+y)=2(a+b) so take c = (a+b) [add equations in 4 and 6] There is an integer, e.g, c, with (x+y)=2c [def. of divisible] [def. of even] [def. of even] • Show that a given statement is true by – With or without mathematical manipulations • Template for Proof of an if-then theorem – First sentence(s) of proof is the hypothesis – Last sentence(s) of proof is the conclusion of – Unwind the definitions, working from both end – Try to forge a ‘link’ between the two halves of

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7 Direct Proof Example (1/2) Given :
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20_proofs - Introduction to Computers and Programming Prof...

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