Proof by contradiction (aka a LaTeX test)

What is a proof by contradiction? A proof by contradiction is if $\neg P \Rightarrow F$ is true.

One assumes that a proposition P is False, and uses that to derive until a contradiction is reached, which can’t be True.

A popular example: Let’s prove that $\sqrt2$ is irrational. An irrational number is something that cannot be expanded into a fraction. (A common misconception is that Pi is $\frac{22}{7}$ and therefore rational; no it is not exactly $\frac{22}{7}$.


Assume that $\sqrt2$ is rational; such that we can represent it as $\frac{a}{b}$, where $\frac{a}{b}$ is a fraction in lowest terms.

$\Rightarrow \sqrt2 = \frac{a}{b}$ $\Rightarrow 2 = \frac{a^2}{b^2}$ $\Rightarrow 2b^2 = a^2$ $\Rightarrow 2 \mid a$ $\Rightarrow 4 \mid a^2$ $\Rightarrow 4 \mid 2b^2$ $\Rightarrow 2 \mid b^2$ $\Rightarrow 2 \mid b$ If both $a$ and $b$ are even, they are not in lowest terms, as both can be divided by 2 for further simplification. Hence we have a contradiction. $\square$


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