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.
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 andDavid 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.
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.
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).
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?
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).
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?
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
Let $P(n)$ be a predicate. To prove $P(n)$ is true for all $n \geq n_0$, we prove:
* **Base case**: Prove $P(n_0)$ is true. This is usually done directly.
* **Inductive case**: Prove $P(k) \Rightarrow P(k + 1)$ for all $k \geq n_0$.
Within the inductive case, $P(k)$ is known as the **inductive hypothesis**.
%%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?
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 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](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
<!--ID: 1714530152713-->
END%%
%%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](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
Strong induction expands the induction hypothesis. Let $P(n)$ be a predicate. To prove $P(n)$ is true for all $n \geq n_0$, we prove:
* **Base case**: Prove $P(n_0)$ is true. This is usually done directly.
* **Inductive case**: Assume $P(k)$ is true for all $n_0 \leq k <n$.Thenprove$P(n)$istrue.
%%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](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
<!--ID: 1714574131949-->
END%%
%%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](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
<!--ID: 1714574131955-->
END%%
%%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](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
<!--ID: 1714574131963-->
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%%
## 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).
* 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.
