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Com S 280 - Induction and Recursion

Com S 280 - Induction and Recursion - Induction and...

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Induction and Recursion - Definitions o sequence: a function with domain that is a subset of the set of integers o geometric progression: a sequence of the form , , , 2 ar ar a , where a and r are real numbers o arithmetic progression: a sequence of the form , 2 , , d a d a a + + , where a and d are real numbers o the principle of mathematical induction: The statement ) ( n nP 2200 is true if ) 1 ( P is true and [ ] ) 1 ( ) ( + 2200 k P k P k is true o basis step: the proof of ) 1 ( P in a proof by mathematical induction of ) ( n nP 2200 o inductive step: the proof of ) 1 ( ) ( + k P k P for all positive integers k in a proof by mathematical induction of ) ( n nP 2200 o strong induction: The statement ) ( n nP 2200 is true if ) 1 ( P and ( 29 [ ] ) 1 ( ) ( ^ )^ 1 ( + 2200 k P k P P k is true o well-ordering property: Every nonempty set of nonnegative integers has a least element o recursive definition of a function: a definition of a function that specifies an initial set of values and a rules for obtaining values of this function at integers from its values at smaller integers. o recursive definition of a set: a definition of a set that specifies an initial set of elements in the set
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