Question Number 178448 by Spillover last updated on 16/Oct/22
$$\mathrm{Using}\:\mathrm{the}\:\mathrm{algebra}\:\mathrm{propositions} \\ $$$$\mathrm{simplify} \\ $$$$\left(\mathrm{p}\leftrightarrow\mathrm{q}\right)\rightarrow\left(\mathrm{p}\rightarrow\mathrm{q}\right) \\ $$
Answered by greougoury555 last updated on 16/Oct/22
![[(p→q)∧(q→p)]→(p→q) = [ (p∧∼q)∨(q∧∼p)]∨(∼p∨q) = (p∧∼q)∨(∼p∧q)∨(∼p∨q) = T](Q178463.png)
$$\left[\left({p}\rightarrow{q}\right)\wedge\left({q}\rightarrow{p}\right)\right]\rightarrow\left({p}\rightarrow{q}\right) \\ $$$$=\:\left[\:\left({p}\wedge\sim{q}\right)\vee\left({q}\wedge\sim{p}\right)\right]\vee\left(\sim{p}\vee{q}\right) \\ $$$$=\:\left({p}\wedge\sim{q}\right)\vee\left(\sim{p}\wedge{q}\right)\vee\left(\sim{p}\vee{q}\right) \\ $$$$=\:{T} \\ $$