notebook/notes/proofs/induction.md

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title TARGET DECK FILE TAGS tags
Induction Obsidian::STEM algebra::sequence proof
proof
sequence

Overview

%%ANKI Cloze The {base case} is to induction whereas {initial conditions} are to recursive definitions. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

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%%ANKI Cloze The {inductive case} is to induction whereas {recurrence relations} are to recursive definitions. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

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%%ANKI Basic What standard names are given to the cases in an induction proof? Back: The base case and inductive case. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

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%%ANKI Basic Let (a_n)_{n \geq 0} = P(n) and P(n) \Leftrightarrow n \geq 2. How is (a_n) written with terms expanded? Back: F, F, T, T, T, \ldots Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

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%%ANKI Basic If proving P(n) by weak induction, what are the first five terms of the underlying sequence? Back: P(0), P(1), P(2), P(3), P(4), \ldots Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

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%%ANKI Basic What proposition is typically proven in the base case of an inductive proof? Back: P(n_0) for some n_0 \geq 0. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

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%%ANKI Basic What proposition is typically proven in the inductive case of an inductive proof? Back: P(k) \Rightarrow P(k + 1) for all k \geq n_0. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

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%%ANKI Basic In weak induction, what special name is given to the antecedent of P(k) \Rightarrow P(k + 1)? Back: The inductive hypothesis. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

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%%ANKI Cloze {Closed formulas} are to recursive definitions as {direct proofs} are to proof strategies. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

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%%ANKI Cloze {Recurrence relations} are to recursive definitions as {induction} is to proof strategies. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

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%%ANKI Basic What proof strategy is most directly tied to recursion? Back: Induction. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

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%%ANKI Basic Using typical identifiers, what is the inductive hypothesis of P(n) using weak induction? Back: Assume P(k) for some k \geq n_0. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

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%%ANKI Basic Using typical identifiers, what is the inductive hypothesis of P(n) using strong induction? Back: Assume P(k) for all k < n. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

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%%ANKI Basic Why is strong induction considered stronger than weak induction? Back: It can be used to solve at least the same set of problems weak induction can. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

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%%ANKI Basic What negation is introduced to explain why the strong induction assumption is valid? Back: If P(n) is not true for all n, there exists a first n_0 for which \neg P(n_0). Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

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%%ANKI Basic What distinguishes the base case of weak and strong induction proofs? Back: The latter may have more than one base case. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

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Bibliography