## Direct proof --- ## Mathematical argument #### Theorem Dave is employed by University of Newcastle. #### Proof Dave lectures MATH2340. Dave is employed by University of Newcastle.   □ <!-- .slide: data-auto-animate --> --- ## Mathematical argument #### Theorem Dave is employed by University of Newcastle. #### Proof Dave lectures MATH2340. Therefore, Dave is employed by University of Newcastle.   □ <!-- .slide: data-auto-animate --> --- ## Mathematical argument #### Theorem Dave is employed by University of Newcastle. #### Proof Dave lectures MATH2340. $\implies$ Dave is employed by University of Newcastle.   □ <!-- .slide: data-auto-animate --> --- ## Mathematical argument #### Theorem Dave is employed by University of Newcastle. #### Proof Dave lectures MATH2340. $\implies$ Dave is employed by University of Newcastle.   □ #### Lesson Without clear implications, there is no argument. <!-- .slide: data-auto-animate --> --- ## Using known facts #### <span class="fragment daz-switch preload" data-fragment-index="0">Theorem (Pythagoras)</span><span class="fragment daz-switch color1" data-fragment-index="1">Theorem (Pythagoras)</span> The side lengths $a$, $b$, $c$ of a right triangle, with $c$ the hypotenuse, satisfy $a^2+b^2=c^2$. #### <span class="fragment daz-switch preload" data-fragment-index="0">Theorem 2</span><span class="fragment daz-switch color2" data-fragment-index="1">Theorem 2</span> Positive real numbers are those which represent lengths. #### <span class="fragment daz-switch preload" data-fragment-index="0">Theorem 3</span><span class="fragment daz-switch color3" data-fragment-index="1">Theorem 3</span> All natural numbers are real numbers. #### Theorem 4 There is a positive real number $z$ such that $z^2=2$. #### Proof As $1\in$ <span class="fragment daz-switch preload" data-fragment-index="0">$\mathbb N\subset\mathbb R$</span><span class="fragment daz-switch color3" data-fragment-index="1">$\mathbb N\subset\mathbb R$</span>, there is a right isosceles triangle whose catheti have <span class="fragment daz-switch preload" data-fragment-index="0">length $1$</span><span class="fragment daz-switch color2" data-fragment-index="1">length $1$</span>. Let $z$ be the length of this triangle's hypotenuse. As $z$ is a length, <span class="fragment daz-switch preload" data-fragment-index="0">$z\in\mathbb R$</span><span class="fragment daz-switch color2" data-fragment-index="1">$z\in\mathbb R$</span>. By Pythagoras's theorem, <span class="fragment daz-switch preload" data-fragment-index="0">$z^2=1^2+1^2$</span><span class="fragment daz-switch color1" data-fragment-index="1">$z^2=1^2+1^2$</span> $=2$.   □ <!-- .slide: class="smallertext" --> --- ### Direction of implications #### Correct Dave lectures MATH2340. $\implies$ Dave works at UoN. #### Incorrect Dave works at UoN. $\implies$ Dave lectures MATH2340. #### Also correct Dave works at UoN. $\impliedby$ Dave lectures MATH2340. #### Also correct $a^2+b^2=0$. $\iff$ $a=0$ and $b=0$. ---