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PMI10F - ⇒ P n 1 is true then P n is true for each n ∈...

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PRINCIPLE OF MATHEMATICAL INDUCTION (PMI) Let N = { 1 , 2 , 3 ,... } be the natural numbers. Let Z = { ..., - 2 , - 1 , 0 , 1 , 2 ,... } be the integers. Let P ( n ) be a statement (that is either true or false) about n . Sometimes we denote P ( n ) by P n . § 2.4: PMI (basic form) If base step: P (1) is true inductive step: for each n N : [ P ( n ) is true ] = [ P ( n + 1) is true ] then P ( n ) is true for each n N . § 2.4: PMI (generalized form) (also known as: doesn’t matter where you start form) Fix n 0 Z . If base step: P ( n 0 ) is true inductive step: for each n Z with n n 0 : [ P ( n ) is true ] =
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Unformatted text preview: ⇒ [ P ( n + 1) is true ] then P ( n ) is true for each n ∈ Z such that n ≥ n . § 2.5: PMI (strong form) (our book calls this PCI = Principle of Complete Induction) Fix n ∈ Z . If base step: P ( n ) is true inductive step: for each n ∈ Z with n ≥ n : [ P ( j ) is true for j = n , 1 + n ,...,n ] = ⇒ [ P ( n + 1) is true ] then P ( n ) is true for each n ∈ Z such that n ≥ n . 1...
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