--- 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). END%% %%ANKI Basic What is PMI an acronym for? Back: The **p**rinciple of **m**athematical **i**nduction. Reference: N/A. END%% %%ANKI Basic What is WOP an acronym for? Back: The **w**ell-**o**rdering **p**rinciple. Reference: N/A. 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.