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Introduction to Proofs
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Introduction A proof is a valid argument establishing the truth of a mathematical statement. We (CS) are interested in proof to establish the correctness of an algorithm the correctness of a program a property of a system (e.g., Java can be parsed efficiently) AI, automated theorem proving the consistency of a specification . . .
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Methods for Proving Theorems
The theorem typically is of the form x, ( P(x) Q(x) ). The proof method typically is of the form Let c be an arbitrary c in the domain.. Show that P(c) Q(c) is true. (The essence.) Therefore, x, ( P(x) Q(x) ).
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Direct Proof of p q Assume that p is true. Show that q is true.
Let the integers, Z, be the domain. Prove that x (( 3 | x – 2 ) ( 3 | x2 – 1 )) 3 | x – 2 (Assumption) x – 2 0 modulo 3. x 2 modulo 3. 4. x2 1 mod 3. x2 – 1 0 mod 3. 3 | x2 – 1.
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Proof by Contraposition of p q
Direct proof of the contrapositive: Assume ~q is true. Show that ~p is true.
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Proof by Contraposition of p q
Example Let the domain be the integers. Prove: (ab is even) (a is even b is even) Assume (a is even b is even). a is odd b is odd. m, a = 2m + 1; n, b = 2n + 1. ab = (2m + 1)(2n + 1) = 4mn + 2m + 2n + 1 = 2(2mn + m + n) + 1 is odd.
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Proof by Contradiction of p q
Show p q is true by showing that (p q) ≡ ( p q) ≡ (p q) is false.
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Example If , 2, …, 10 are placed randomly in a circle then the sum of some 3 adjacent numbers 17. 1. Assume that 1, 2, …, 10 are placed randomly in a circle and that the sum of no 3 adjacent numbers 17. 2. Then x1 + x2 + x3 16 x2 + x3 + x4 16 … x10 + x1 + x2 16. 3. Summing, 3( x1 + x x10 ) 16 10 = 160. 4. But, 3 (x1 + x x10 ) = 3( … + 10) = 3 55 = 165, a contradiction.
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Proving p q p q means p if and only if q p if q means q p
p only if q means if q then p, which means p q. So, p q means p q q p. To prove p q it thus is necessary and sufficient to show that p q q p.
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Characters ≥ ≡
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