Question Number 46475 by peter frank last updated on 27/Oct/18
Commented by Joel578 last updated on 27/Oct/18
$${pls}\:{recheck}\:{question}\:\left({b}\right) \\ $$
Commented by peter frank last updated on 28/Oct/18
![b)[(p→q)∧(q⇔p)∧∼p]→∼q](https://www.tinkutara.com/question/Q46554.png)
$$\left.\mathrm{b}\right)\left[\left(\mathrm{p}\rightarrow\mathrm{q}\right)\wedge\left(\mathrm{q}\Leftrightarrow\mathrm{p}\right)\wedge\sim\mathrm{p}\right]\rightarrow\sim\mathrm{q} \\ $$
Answered by Joel578 last updated on 27/Oct/18
![(p → q) ∧ (q → ∼r) ∧ r ≡ (∼p ∨ q) ∧ [(∼q ∨ ∼r) ∧ r] ≡ (∼p ∨ q) ∧ [(r ∧ ∼q) ∨ (r ∧ ∼r)] ≡ (∼p ∨ q) ∧ [(r ∧ ∼q) ∨ F] ≡ (∼p ∨ q) ∧ (r ∧ ∼q) ≡ [(∼p ∨ q) ∧ ∼q] ∧ r ≡ [(∼q ∧ ∼p) ∨ (∼q ∧ q)] ∧ r ≡ [(∼q ∧ ∼p) ∨ F] ∧ r ≡ (∼q ∧ ∼p) ∧ r [(∼q ∧ ∼p) ∧ r] → ∼p ≡ ∼[(∼q ∧ ∼p) ∧ r] ∨ ∼p ≡ [(q ∨ p) ∨ ∼r] ∨ ∼p ≡ q ∨ (p ∨ ∼p) ∨ ∼r ≡ q ∨ T ∨ ∼r ≡ T](https://www.tinkutara.com/question/Q46504.png)
$$\left({p}\:\rightarrow\:{q}\right)\:\wedge\:\left({q}\:\rightarrow\:\sim{r}\right)\:\wedge\:{r} \\ $$$$\equiv\:\left(\sim{p}\:\vee\:{q}\right)\:\wedge\:\left[\left(\sim{q}\:\vee\:\sim{r}\right)\:\wedge\:{r}\right] \\ $$$$\equiv\:\left(\sim{p}\:\vee\:{q}\right)\:\wedge\:\left[\left({r}\:\wedge\:\sim{q}\right)\:\vee\:\left({r}\:\wedge\:\sim{r}\right)\right] \\ $$$$\equiv\:\left(\sim{p}\:\vee\:{q}\right)\:\wedge\:\left[\left({r}\:\wedge\:\sim{q}\right)\:\vee\:{F}\right] \\ $$$$\equiv\:\left(\sim{p}\:\vee\:{q}\right)\:\wedge\:\left({r}\:\wedge\:\sim{q}\right) \\ $$$$\equiv\:\left[\left(\sim{p}\:\vee\:{q}\right)\:\wedge\:\sim{q}\right]\:\wedge\:{r} \\ $$$$\equiv\:\left[\left(\sim{q}\:\wedge\:\sim{p}\right)\:\vee\:\left(\sim{q}\:\wedge\:{q}\right)\right]\:\wedge\:{r} \\ $$$$\equiv\:\left[\left(\sim{q}\:\wedge\:\sim{p}\right)\:\vee\:{F}\right]\:\wedge\:{r} \\ $$$$\equiv\:\left(\sim{q}\:\wedge\:\sim{p}\right)\:\wedge\:{r} \\ $$$$ \\ $$$$\left[\left(\sim{q}\:\wedge\:\sim{p}\right)\:\wedge\:{r}\right]\:\rightarrow\:\sim{p} \\ $$$$\equiv\:\sim\left[\left(\sim{q}\:\wedge\:\sim{p}\right)\:\wedge\:{r}\right]\:\vee\:\sim{p} \\ $$$$\equiv\:\left[\left({q}\:\vee\:{p}\right)\:\vee\:\sim{r}\right]\:\vee\:\sim{p} \\ $$$$\equiv\:{q}\:\vee\:\left({p}\:\vee\:\sim{p}\right)\:\vee\:\sim{r} \\ $$$$\equiv\:{q}\:\vee\:{T}\:\vee\:\sim{r} \\ $$$$\equiv\:{T} \\ $$
Commented by peter frank last updated on 28/Oct/18
$$\mathrm{thanks} \\ $$