Question Number 79974 by TawaTawa last updated on 29/Jan/20
Answered by Rio Michael last updated on 30/Jan/20
![solution [(p ∨ q) ∧ (∼p ∨r)] ⇒ (q ∨r) we know from known facts that −(p ∨q) is a contingency (neither a tautology nor contradiction) − (∼p ∨r) is a contingency so (p ∨ q) ∧(∼p ∨r) ⇒ contingency. When ever we use an implication (⇒) for two contingency′s the result is a tautology. But (q ∨r )= contingency therefore [(p ∨q) ∨(∼p ∨r)] ⇒ (q ∨r) is a tautology. formally. [(p ∨q) ∧(∼p ∨r)] ⇒ (q ∨r) = [ (p ∨ q) ∧ ∼(p ∧r)] ⇒(q ∨r) = ∼ [ (∼p ∧∼q) ∨(p ∧r) ⇐(∼q ∨∼r)] = ∼( contradiction) = tautology](https://www.tinkutara.com/question/Q80066.png)
$$\:\boldsymbol{\mathrm{solution}} \\ $$$$\:\:\left[\left({p}\:\vee\:{q}\right)\:\wedge\:\left(\sim{p}\:\vee{r}\right)\right]\:\Rightarrow\:\left({q}\:\vee{r}\right) \\ $$$$\mathrm{we}\:\mathrm{know}\:\mathrm{from}\:\mathrm{known}\:\mathrm{facts}\:\mathrm{that}\: \\ $$$$\:−\left({p}\:\vee{q}\right)\:\mathrm{is}\:\mathrm{a}\:\mathrm{contingency}\:\left(\mathrm{neither}\:\mathrm{a}\:\mathrm{tautology}\:\mathrm{nor}\:\mathrm{contradiction}\right) \\ $$$$\:−\:\left(\sim{p}\:\vee{r}\right)\:\mathrm{is}\:\mathrm{a}\:\mathrm{contingency} \\ $$$$\mathrm{so}\:\:\left(\mathrm{p}\:\vee\:{q}\right)\:\wedge\left(\sim{p}\:\vee{r}\right)\:\Rightarrow\:\mathrm{contingency}. \\ $$$$\mathrm{When}\:\mathrm{ever}\:\mathrm{we}\:\mathrm{use}\:\mathrm{an}\:\mathrm{implication}\:\left(\Rightarrow\right)\:\mathrm{for}\:\mathrm{two}\: \\ $$$$\mathrm{contingency}'\mathrm{s}\:\mathrm{the}\:\mathrm{result}\:\mathrm{is}\:\mathrm{a}\:\mathrm{tautology}. \\ $$$$\mathrm{But}\:\:\left({q}\:\vee{r}\:\right)=\:{contingency} \\ $$$${therefore}\:\:\left[\left({p}\:\vee{q}\right)\:\vee\left(\sim{p}\:\vee{r}\right)\right]\:\Rightarrow\:\left({q}\:\vee{r}\right)\:{is}\:{a}\:{tautology}. \\ $$$$\boldsymbol{\mathrm{formally}}. \\ $$$$\:\left[\left({p}\:\vee{q}\right)\:\wedge\left(\sim{p}\:\vee{r}\right)\right]\:\Rightarrow\:\left({q}\:\vee{r}\right)\:=\:\:\left[\:\left({p}\:\vee\:{q}\right)\:\wedge\:\sim\left({p}\:\wedge{r}\right)\right]\:\Rightarrow\left({q}\:\vee{r}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\sim\:\left[\:\left(\sim{p}\:\wedge\sim{q}\right)\:\vee\left({p}\:\wedge{r}\right)\:\Leftarrow\left(\sim{q}\:\vee\sim{r}\right)\right] \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\sim\left(\:{contradiction}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:{tautology} \\ $$$$ \\ $$
Commented by TawaTawa last updated on 03/Feb/20
$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir} \\ $$