notebook/notes/proofs/index.md

title	TARGET DECK	FILE TAGS	tags
Proofs	Obsidian::STEM	proof::method	proof

Overview

A direct proof is a sequence of statements, either givens or deductions of previous statements, whose last statement is the conclusion to be proved.

%%ANKI Basic What is a direct proof? Back: A proof whose arguments follow directly one after another, up to the conclusion. Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.

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%%ANKI Basic Generally speaking, what should the first statement of a direct proof be? Back: A hypothesis, if one exists. Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.

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%%ANKI Basic Generally speaking, what should the last statement of a direct proof be? Back: The conclusion to be proved. Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.

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An indirect proof works by assuming the denial of the desired conclusion leads to a contradiction in some way.

%%ANKI Basic What is an indirect proof? Back: A proof in which the denial of a proposition is assumed and shown to yield a contradiction. Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.

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%%ANKI Cloze A {direct} proof is contrasted to an {indirect} proof. Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.

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Conditional Proofs

A conditional proof is a proof method used to prove a conditional statement, i.e. statements of form: $P_1 \land \cdots \land P_n \Rightarrow Q$ Note we can assume all the hypotheses are true since if one were false, the implication holds regardless. Direct proofs of the above form are called conditional proofs (CP).

%%ANKI Basic What are conditional proofs? Back: Methods for proving propositions of form P_1 \land \cdots \land P_n \Rightarrow Q. Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.

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%%ANKI Basic Which of conditional proofs or direct proofs is more general? Back: N/A. Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.

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%%ANKI Basic Which of conditional proofs or indirect proofs is more general? Back: N/A. Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.

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%%ANKI Basic Conditional proofs are used to solve propositions of what form? Back: P_1 \land \cdots \land P_n \Rightarrow Q Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.

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%%ANKI Basic How do we justify assuming the hypotheses in a conditional proof? Back: If any hypothesis were false, the conditional we are proving trivially holds. Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.

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%%ANKI Basic Which proof method does CP stand for? Back: Conditional proofs. Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.

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%%ANKI Basic Which natural deduction rule depends directly on the existence of a conditional proof? Back: {\Rightarrow}{\text{-}}I Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

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Proof by Contraposition

Since a conditional and its contrapositive are logically equivalent, we can instead prove the negation of the conclusion leads to the negation of our hypotheses.

%%ANKI Cloze {P \Rightarrow Q} is the contrapositive of {\neg Q \Rightarrow \neg P}. Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.

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%%ANKI Basic Consider conditional P \Rightarrow Q. A proof by contrapositive typically starts with what assumption? Back: \neg Q Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.

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%%ANKI Basic How do you perform a proof by contraposition? Back: By showing the negation of the conclusion yields the negation of the hypotheses. Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.

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%%ANKI Basic Why is proof by contraposition valid? Back: A conditional and its contrapositive are logically equivalent. Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.

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%%ANKI Basic Is a proof by contraposition considered direct or indirect? Back: Indirect. Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.

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Proof by Contradiction

To prove a proposition P by contradiction, we assume \neg P and derive a statement known to be false. Since mathematics is (in most cases) consistent, P must be true.

%%ANKI Basic Consider conditional P \Rightarrow Q. A proof by contradiction typically starts with what assumption? Back: \neg P Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.

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%%ANKI Basic What are the two most common indirect conditional proof strategies? Back: Proof by contraposition and proof by contradiction. Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.

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%%ANKI Basic How do you perform a proof by contradiction? Back: Assume the negation of some statement and derive a contradiction. Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.

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%%ANKI Basic Why is proof by contradiction valid? Back: It's assumed mathematics is consistent. If we prove a false statement, then our assumption is wrong. Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.

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%%ANKI Basic Is a proof by contradiction considered direct or indirect? Back: Indirect. Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.

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%%ANKI Basic Which natural deduction inference rules embody proof by contradiction? Back: \neg{\text{-}}I and \neg{\text{-}}E. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

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Existence Proofs

An existence proof is a proof method used to prove an existential statement, i.e. statements of form: \exists x, P(x)

%%ANKI Basic What are existence proofs? Back: Methods for proving propositions of form \exists x, P(x). Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.

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%%ANKI Basic Which of existence proofs or direct proofs is more general? Back: N/A. Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.

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%%ANKI Basic Which of existence proofs or indirect proofs is more general? Back: N/A. Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.

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%%ANKI Basic Existence proofs are used to solve propositions of what form? Back: \exists x, P(x) Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.

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An existence proof is said to be constructive if it demonstrates the existence of an object by creating (or providing a method for creating) the object. Otherwise it is said to be non-constructive.

%%ANKI Basic Which more general proof method do constructive proofs fall under? Back: Existence proofs. Reference: “Constructive Proof,” in Wikipedia, April 4, 2024, https://en.wikipedia.org/w/index.php?title=Constructive_proof.

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%%ANKI Basic Which of existence proofs or constructive proofs is more general? Back: Existence proofs. Reference: “Constructive Proof,” in Wikipedia, April 4, 2024, https://en.wikipedia.org/w/index.php?title=Constructive_proof.

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%%ANKI Basic Is a constructive proof usually direct or indirect? Back: Usually direct. Reference: “Constructive Proof,” in Wikipedia, April 4, 2024, https://en.wikipedia.org/w/index.php?title=Constructive_proof.

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%%ANKI Basic Which more general proof method do non-constructive proofs fall under? Back: Existence proofs. Reference: “Constructive Proof,” in Wikipedia, April 4, 2024, https://en.wikipedia.org/w/index.php?title=Constructive_proof.

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%%ANKI Basic Which of non-constructive proofs or existence proofs is more general? Back: Existence proofs. Reference: “Constructive Proof,” in Wikipedia, April 4, 2024, https://en.wikipedia.org/w/index.php?title=Constructive_proof.

END%%

%%ANKI Basic Is a non-constructive proof usually direct or indirect? Back: Usually indirect. Reference: “Constructive Proof,” in Wikipedia, April 4, 2024, https://en.wikipedia.org/w/index.php?title=Constructive_proof.

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Induction

Weak Induction

Let P(n) be a predicate depending on a number n \in \mathbb{N}. Assume that

• Base case: P(n_0) is true for some n_0 \geq 0, and
• Inductive case: for all k \geq n_0, P(k) \Rightarrow P(k + 1).

Then P(n) is true for all n \geq n_0.

Within the inductive case, P(k) is known as the inductive hypothesis. The formal justification of proof by induction is intimately tied to the idea of natural-numbers#Inductive Sets.

%%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 a weak induction 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 a weak induction 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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Strong Induction

Let P(n) be a predicate depending on a number n \in \mathbb{N}. Assume that

• Base case: P(n_0) is true for some n_0 \geq 0, and
• Inductive case: for all k \geq n_0, P(n_0) \land P(n_0 + 1) \land \cdots \land P(k) \Rightarrow P(k + 1).

Then P(n) is true for all n \geq n_0.

The formal justification of proof by induction is intimately tied to the idea of natural-numbers#Inductive Sets and the natural-numbers#Well-Ordering Principle.

%%ANKI Basic Using typical identifiers, what is the inductive hypothesis of P(n) using strong induction? Back: Assume P(k) for all n_0 \leq 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 makes strong induction "stronger" than weak induction? Back: It gives more propositions in the antecedent of the 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 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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%%ANKI Basic How is the following strong induction clause rewritten to use weak induction? $P(0) \land P(1) \land \cdots \land P(k) \Rightarrow P(k + 1)$ Back: As Q(k) \Rightarrow Q(k + 1) where Q(n) = P(0) \land P(1) \land \cdots \land P(n) for all n \in \omega. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).

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%%ANKI Basic How is the following weak induction clause rewritten to use strong induction? $P(k) \Rightarrow P(k + 1)$ Back: As P(n_0) \land P(n_0 + 1) \land \cdots \land P(k) \Rightarrow P(k + 1) for some 0 \leq n_0. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).

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Well-Ordering Principle

This is covered natural-numbers#Well-Ordering Principle. It is equivalent to weak and strong induction.

%%ANKI Basic What are the three most commonly used principles of induction? Back: Weak induction, strong induction, and well-ordering. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).

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%%ANKI Basic Why are names "weak" and "strong" induction a misnomer? Back: Weak and strong induction are logically equivalent. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).

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%%ANKI Basic What is PMI an acronym for? Back: The principle of mathematical induction. Reference: N/A.

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%%ANKI Basic What is WOP an acronym for? Back: The well-ordering principle. Reference: N/A.

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Bibliography

• “Constructive Proof,” in Wikipedia, April 4, 2024, https://en.wikipedia.org/w/index.php?title=Constructive_proof.
• Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
• Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
• Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.