induction - Introduction to the Theory of Computation...

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Introduction to the Theory of Computation AZADEH FARZAN WINTER 2010 Monday, January 11, 2010
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PROOFS Proof by Contradiction Proof by Construction Proof by Induction Jack sees Jill, who has just come in from outdoors dry. Jack knows that it is not raining. His proof : if it were raining ( assuming the statement is false ), Jill woud be wet ( contradiction, or false consequence ). Therefore, it must not be raining. For any n, there exist n consecutive composite integers. Proof : here are the n consecutive composite integers for any n: (n+1)!+2, (n+1)!+3, ..., (n+1)!+n+1 Monday, January 11, 2010
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INDUCTION A method used to show that all elements of an infinite set have a specific property. Example: an arithmetic expression computes a desired quantity for every assignment to its variables. Example: program works correctly at all steps or for all input. Every proof induction consists of two parts: The basis . The induction step . Monday, January 11, 2010
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IDEA Let’s take the infinite set to be N = { 1 , 2 , 3 , . . . } . Let’s call the property P . The goal is to prove that P ( k ) is true for each natural number k . The basis: P (1) is true. The induction step: if P ( i ) is true, then P ( i + 1) is true as well. P ( 1 ) P ( 2 ) P ( 3 ) . . . P ( i ) . . .
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