:PROPERTIES: :ID: 4ed986aa-c558-4302-b9b5-e8a67d4b84fb :mtime: 20231103034628 :ctime: 20231103034626 :END: #+title: logical implication #+filetags: :public:project: * Rules of implication ** Modus Ponendo Punens \[mpp : P \to (P \implies Q) \to Q \] ** Hypothetic Syllogism * How to prove logical implication ** Proof template In order to prove that \[P \implies Q\] Let us first assume $P$. Then, we want to show $Q$. ** Examples of logical implication proofs *** Example : $\sqrt{x} rational \implies x rational$ Let us assume that $\sqrt{x}$ is rational. We want to show that $x$ is rational. By the definition of rational, there exist integers $m,n$ such that \[\sqrt{x} = \frac{m}{n}\] If we square both sides, we get \[x = \frac{m^{2}}{n^{2}}\] Since $m^2$ and $n^2$ are integers, then $x$ is in the form of a rational number, so we are done.