138 lines
6.1 KiB
Markdown
138 lines
6.1 KiB
Markdown
---
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title: Induction
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TARGET DECK: Obsidian::STEM
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FILE TAGS: algebra::sequence proof
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tags:
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- proof
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- sequence
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---
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## Overview
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%%ANKI
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Cloze
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The {base case} is to induction whereas {initial conditions} are to recursive definitions.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1714530152689-->
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END%%
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%%ANKI
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Cloze
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The {inductive case} is to induction whereas {recurrence relations} are to recursive definitions.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1714530152697-->
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END%%
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%%ANKI
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Basic
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What standard names are given to the cases in an induction proof?
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Back: The base case and inductive case.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1714530152701-->
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END%%
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%%ANKI
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Basic
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Let $(a_n)_{n \geq 0} = P(n)$ and $P(n) \Leftrightarrow n \geq 2$. How is $(a_n)$ written with terms expanded?
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Back: $F$, $F$, $T$, $T$, $T$, $\ldots$
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1714530152705-->
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END%%
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%%ANKI
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Basic
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If proving $P(n)$ by weak induction, what are the first five terms of the underlying sequence?
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Back: $P(0)$, $P(1)$, $P(2)$, $P(3)$, $P(4)$, $\ldots$
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1714530152709-->
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END%%
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%%ANKI
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Basic
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What proposition is typically proven in the base case of an inductive proof?
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Back: $P(n_0)$ for some $n_0 \geq 0$.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1714530152713-->
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END%%
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%%ANKI
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Basic
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What proposition is typically proven in the inductive case of an inductive proof?
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Back: $P(k) \Rightarrow P(k + 1)$ for all $k \geq n_0$.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1714530152718-->
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END%%
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%%ANKI
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Basic
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In weak induction, what special name is given to the antecedent of $P(k) \Rightarrow P(k + 1)$?
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Back: The inductive hypothesis.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1714530152722-->
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END%%
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%%ANKI
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Cloze
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{Closed formulas} are to recursive definitions as {direct proofs} are to proof strategies.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1714532476735-->
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END%%
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%%ANKI
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Cloze
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{Recurrence relations} are to recursive definitions as {induction} is to proof strategies.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1714532476742-->
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END%%
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%%ANKI
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Basic
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What proof strategy is most directly tied to recursion?
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Back: Induction.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1714574131911-->
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END%%
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%%ANKI
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Basic
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Using typical identifiers, what is the inductive hypothesis of $P(n)$ using weak induction?
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Back: Assume $P(k)$ for some $k \geq n_0$.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1714574131942-->
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END%%
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%%ANKI
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Basic
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Using typical identifiers, what is the inductive hypothesis of $P(n)$ using strong induction?
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Back: Assume $P(k)$ for all $k < n$.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1714574131949-->
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END%%
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%%ANKI
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Basic
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Why is strong induction considered stronger than weak induction?
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Back: It can be used to solve at least the same set of problems weak induction can.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1714574131955-->
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END%%
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%%ANKI
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Basic
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What negation is introduced to explain why the strong induction assumption is valid?
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Back: If $P(n)$ is not true for all $n$, there exists a *first* $n_0$ for which $\neg P(n_0)$.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1714574131963-->
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END%%
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%%ANKI
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Basic
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What distinguishes the base case of weak and strong induction proofs?
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Back: The latter may have more than one base case.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1714574131969-->
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END%%
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## Bibliography
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* Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). |