r
/
notebook
1
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity
notebook/notes/proofs/index.md

493 lines
18 KiB
Markdown
Raw Blame History

This file contains invisible Unicode characters!

This file contains invisible Unicode characters that may be processed differently from what appears below. If your use case is intentional and legitimate, you can safely ignore this warning. Use the Escape button to reveal hidden characters.

---
title: Proofs
TARGET DECK: Obsidian::STEM
FILE TAGS: proof::method
tags:
- 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.
<!--ID: 1721824073057-->
END%%
%%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.
<!--ID: 1721824073062-->
END%%
%%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.
<!--ID: 1721824073065-->
END%%
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.
<!--ID: 1721824073070-->
END%%
%%ANKI
Cloze
A {direct} proof is contrasted to an {indirect} proof.
Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.
<!--ID: 1721824073073-->
END%%
## 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.
<!--ID: 1721824073076-->
END%%
%%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.
<!--ID: 1721824073079-->
END%%
%%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.
<!--ID: 1721824073082-->
END%%
%%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.
<!--ID: 1721824073086-->
END%%
%%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.
<!--ID: 1721824073089-->
END%%
%%ANKI
Basic
Which proof method does CP stand for?
Back: **C**onditional **p**roofs.
Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.
<!--ID: 1721824073092-->
END%%
%%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.
<!--ID: 1721825479299-->
END%%
### 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.
<!--ID: 1721824073095-->
END%%
%%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.
<!--ID: 1721824073098-->
END%%
%%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.
<!--ID: 1721824073101-->
END%%
%%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.
<!--ID: 1721824073104-->
END%%
%%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.
<!--ID: 1721824073108-->
END%%
### 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.
<!--ID: 1721824073112-->
END%%
%%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.
<!--ID: 1721824073116-->
END%%
%%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.
<!--ID: 1721824073121-->
END%%
%%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.
<!--ID: 1721824073125-->
END%%
%%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.
<!--ID: 1721824073130-->
END%%
%%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.
<!--ID: 1721825479310-->
END%%
## 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.
<!--ID: 1721824073134-->
END%%
%%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.
<!--ID: 1721824073137-->
END%%
%%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.
<!--ID: 1721824073140-->
END%%
%%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.
<!--ID: 1721824073143-->
END%%
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](https://en.wikipedia.org/w/index.php?title=Constructive_proof&oldid=1217198357).
<!--ID: 1721824073146-->
END%%
%%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](https://en.wikipedia.org/w/index.php?title=Constructive_proof&oldid=1217198357).
<!--ID: 1722336217056-->
END%%
%%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](https://en.wikipedia.org/w/index.php?title=Constructive_proof&oldid=1217198357).
<!--ID: 1721824073149-->
END%%
%%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](https://en.wikipedia.org/w/index.php?title=Constructive_proof&oldid=1217198357).
<!--ID: 1721824073152-->
END%%
%%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](https://en.wikipedia.org/w/index.php?title=Constructive_proof&oldid=1217198357).
<!--ID: 1722336217060-->
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](https://en.wikipedia.org/w/index.php?title=Constructive_proof&oldid=1217198357).
<!--ID: 1721824073155-->
END%%
## 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|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](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
<!--ID: 1714530152689-->
END%%
%%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](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
<!--ID: 1714530152697-->
END%%
%%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](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
<!--ID: 1714530152701-->
END%%
%%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](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
<!--ID: 1714530152705-->
END%%
%%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](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
<!--ID: 1714530152709-->
END%%
%%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](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
<!--ID: 1714530152713-->
END%%
%%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](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
<!--ID: 1714530152718-->
END%%
%%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](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
<!--ID: 1714530152722-->
END%%
%%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](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
<!--ID: 1714532476735-->
END%%
%%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](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
<!--ID: 1714532476742-->
END%%
%%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](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
<!--ID: 1714574131911-->
END%%
%%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](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
<!--ID: 1714574131942-->
END%%
### 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|inductive sets]] and the [[natural-numbers#Well-Ordering Principle|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](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
<!--ID: 1714574131949-->
END%%
%%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](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
<!--ID: 1714574131955-->
END%%
%%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](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
<!--ID: 1714574131969-->
END%%
%%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).
<!--ID: 1731203636959-->
END%%
%%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).
<!--ID: 1731203636963-->
END%%
### Well-Ordering Principle
This is covered [[natural-numbers#Well-Ordering Principle|here]]. 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).
<!--ID: 1731203636955-->
END%%
%%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).
<!--ID: 1731204485580-->
END%%
%%ANKI
Basic
What is PMI an acronym for?
Back: The **p**rinciple of **m**athematical **i**nduction.
Reference: N/A.
<!--ID: 1731205303107-->
END%%
%%ANKI
Basic
What is WOP an acronym for?
Back: The **w**ell-**o**rdering **p**rinciple.
Reference: N/A.
<!--ID: 1731205303114-->
END%%
## Bibliography
* “Constructive Proof,” in _Wikipedia_, April 4, 2024, [https://en.wikipedia.org/w/index.php?title=Constructive_proof](https://en.wikipedia.org/w/index.php?title=Constructive_proof&oldid=1217198357).
* 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](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
* Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.
Reference in New Issue View Git Blame Copy Permalink