diff --git a/_proof-writing.qmd b/_proof-writing.qmd index ef7c4aff8..f0310c5e1 100644 --- a/_proof-writing.qmd +++ b/_proof-writing.qmd @@ -92,6 +92,38 @@ The symbols for [logical entailment](#def-logical-entailment) are listed under {{< slidebreak >}} +{{< include mds/_subfiles/proof-writing/_def-tautology.qmd >}} + +{{< slidebreak >}} + +{{< include mds/_subfiles/proof-writing/_exm-tautology.qmd >}} + +{{< slidebreak >}} + +{{< include mds/_subfiles/proof-writing/_def-contradiction.qmd >}} + +{{< slidebreak >}} + +{{< include mds/_subfiles/proof-writing/_exm-contradiction.qmd >}} + +{{< slidebreak >}} + +{{< include mds/_subfiles/proof-writing/_thm-equivalence-iff.qmd >}} + +{{< slidebreak >}} + +{{< include mds/_subfiles/proof-writing/_exm-equivalence-iff.qmd >}} + +{{< slidebreak >}} + +{{< include mds/_subfiles/proof-writing/_thm-double-negation.qmd >}} + +{{< slidebreak >}} + +{{< include mds/_subfiles/proof-writing/_exm-double-negation.qmd >}} + +{{< slidebreak >}} + {{< include mds/_subfiles/proof-writing/_def-quantifier.qmd >}} {{< slidebreak >}} @@ -269,6 +301,10 @@ Let's save $\rightarrow$ (`\rightarrow`) for [convergence](algebra.qmd#def-conve {{< slidebreak >}} +{{< include mds/_subfiles/proof-writing/_thm-contradiction-works.qmd >}} + +{{< slidebreak >}} + {{< include mds/_subfiles/proof-writing/_exm-proof-by-contradiction-truth-table.qmd >}} diff --git a/_subfiles/proof-writing/_def-contradiction.qmd b/_subfiles/proof-writing/_def-contradiction.qmd new file mode 100644 index 000000000..56fe32b76 --- /dev/null +++ b/_subfiles/proof-writing/_def-contradiction.qmd @@ -0,0 +1,7 @@ +:::{#def-contradiction} +#### Contradiction + +A **contradiction** is a [proposition](#def-proposition) +that is false in every row of its [truth table](#def-truth-table). + +::: diff --git a/_subfiles/proof-writing/_def-tautology.qmd b/_subfiles/proof-writing/_def-tautology.qmd new file mode 100644 index 000000000..cf5767cc2 --- /dev/null +++ b/_subfiles/proof-writing/_def-tautology.qmd @@ -0,0 +1,7 @@ +:::{#def-tautology} +#### Tautology + +A **tautology** is a [proposition](#def-proposition) +that is true in every row of its [truth table](#def-truth-table). + +::: diff --git a/_subfiles/proof-writing/_exm-contradiction.qmd b/_subfiles/proof-writing/_exm-contradiction.qmd new file mode 100644 index 000000000..09d72269a --- /dev/null +++ b/_subfiles/proof-writing/_exm-contradiction.qmd @@ -0,0 +1,16 @@ +:::{#exm-contradiction} +#### $P \land \neg P$ is a contradiction + + +By the definitions of [and](#def-and) and [not](#def-not), +the [truth table](#def-truth-table) of $P \land \neg P$ is: + +| $P$ | $\neg P$ | $P \land \neg P$ | +|:---:|:--------:|:----------------:| +| T | F | F | +| F | T | F | + +The last column is false in every row, +so $P \land \neg P$ is a [contradiction](#def-contradiction). + +::: diff --git a/_subfiles/proof-writing/_exm-double-negation.qmd b/_subfiles/proof-writing/_exm-double-negation.qmd new file mode 100644 index 000000000..327a1de38 --- /dev/null +++ b/_subfiles/proof-writing/_exm-double-negation.qmd @@ -0,0 +1,11 @@ +:::{#exm-double-negation} +#### Removing a double negation + + +Let $P$ be the [proposition](#def-proposition) "$7$ is [odd](number-systems.qmd#def-odd)". +Then $\neg \neg P$ is "it is not the case that $7$ is not odd". +By [double negation](#thm-double-negation), +$\neg \neg P$ says the same thing as $P$: +$7$ is odd. + +::: diff --git a/_subfiles/proof-writing/_exm-equivalence-iff.qmd b/_subfiles/proof-writing/_exm-equivalence-iff.qmd new file mode 100644 index 000000000..7075aa51f --- /dev/null +++ b/_subfiles/proof-writing/_exm-equivalence-iff.qmd @@ -0,0 +1,21 @@ +:::{#exm-equivalence-iff} +#### A tautological biconditional + + +@exm-logically-equivalent showed that +$P \Rightarrow Q$ and $\neg P \lor Q$ are [logically equivalent](#def-logically-equivalent). +Adding a column for the biconditional to its [truth table](#def-truth-table) +gives: + +| $P$ | $Q$ | $\neg P \lor Q$ | $P \Rightarrow Q$ | $(P \Rightarrow Q) \Leftrightarrow (\neg P \lor Q)$ | +|:---:|:---:|:---------------:|:-----------------:|:---------------------------------------------------:| +| T | T | T | T | T | +| T | F | F | F | T | +| F | T | T | T | T | +| F | F | T | T | T | + +The last column is true in every row, +so the biconditional is a [tautology](#def-tautology), +as [equivalence as a tautological biconditional](#thm-equivalence-iff) says it must be. + +::: diff --git a/_subfiles/proof-writing/_exm-tautology.qmd b/_subfiles/proof-writing/_exm-tautology.qmd new file mode 100644 index 000000000..ec4ffa4f1 --- /dev/null +++ b/_subfiles/proof-writing/_exm-tautology.qmd @@ -0,0 +1,16 @@ +:::{#exm-tautology} +#### $P \lor \neg P$ is a tautology + + +By the definitions of [or](#def-or) and [not](#def-not), +the [truth table](#def-truth-table) of $P \lor \neg P$ is: + +| $P$ | $\neg P$ | $P \lor \neg P$ | +|:---:|:--------:|:---------------:| +| T | F | T | +| F | T | T | + +The last column is true in every row, +so $P \lor \neg P$ is a [tautology](#def-tautology). + +::: diff --git a/_subfiles/proof-writing/_thm-contradiction-works.qmd b/_subfiles/proof-writing/_thm-contradiction-works.qmd new file mode 100644 index 000000000..1e2744f7b --- /dev/null +++ b/_subfiles/proof-writing/_thm-contradiction-works.qmd @@ -0,0 +1,36 @@ +:::{#thm-contradiction-works} +#### Why a proof by contradiction works + +For [propositions](#def-proposition) $P$ and $C$, + +$$ +P \equiv \paren{\neg P \Rightarrow (C \land \neg C)}. +$$ + +::: + +::: {.content-hidden when-format="revealjs"} + +::: proof + +By the definitions of [and](#def-and) and [not](#def-not), +$C \land \neg C$ is false in every row, +so it is a [contradiction](#def-contradiction). +So, by the definition of [implies](#def-implies), +$\neg P \Rightarrow (C \land \neg C)$ is true exactly in the rows +where $\neg P$ is false, +which are the rows where $P$ is true: + +| $P$ | $C$ | $\neg P$ | $C \land \neg C$ | $\neg P \Rightarrow (C \land \neg C)$ | +|:---:|:---:|:---:|:---:|:---:| +| T | T | F | F | T | +| T | F | F | F | T | +| F | T | T | F | F | +| F | F | T | F | F | + +The first and last columns match in every row, +so the two are [logically equivalent](#def-logically-equivalent). + +::: + +::: diff --git a/_subfiles/proof-writing/_thm-double-negation.qmd b/_subfiles/proof-writing/_thm-double-negation.qmd new file mode 100644 index 000000000..9fdbd081c --- /dev/null +++ b/_subfiles/proof-writing/_thm-double-negation.qmd @@ -0,0 +1,28 @@ +:::{#thm-double-negation} +#### Double negation + +For a [proposition](#def-proposition) $P$, + +$$ +\neg \neg P \equiv P. +$$ + +::: + +::: {.content-hidden when-format="revealjs"} + +::: proof + +By the definition of [not](#def-not), the [truth table](#def-truth-table) is: + +| $P$ | $\neg P$ | $\neg \neg P$ | +|:---:|:---:|:---:| +| T | F | T | +| F | T | F | + +The first and last columns match in both rows, +so the two sides are [logically equivalent](#def-logically-equivalent). + +::: + +::: diff --git a/_subfiles/proof-writing/_thm-equivalence-iff.qmd b/_subfiles/proof-writing/_thm-equivalence-iff.qmd new file mode 100644 index 000000000..b2b4c25da --- /dev/null +++ b/_subfiles/proof-writing/_thm-equivalence-iff.qmd @@ -0,0 +1,23 @@ +:::{#thm-equivalence-iff} +#### Equivalence is a tautological biconditional + +For [propositions](#def-proposition) $A$ and $B$, +$A$ and $B$ are [logically equivalent](#def-logically-equivalent) +exactly when $A \Leftrightarrow B$ is a [tautology](#def-tautology). + +::: + +::: {.content-hidden when-format="revealjs"} + +::: proof + +By the definition of [if and only if](#def-iff), +$A \Leftrightarrow B$ is true in a row of the [truth table](#def-truth-table) +exactly when $A$ and $B$ have the same truth value in that row. +So $A \Leftrightarrow B$ is true in every row +exactly when $A$ and $B$ have the same truth value in every row, +which is the definition of $A \equiv B$. + +::: + +:::