if-p-and-q-are-two-afirmation-p-q-p-q-p-q-

Question Number 1689 by 123456 last updated on 31/Aug/15

$$\mathrm{if}\:{p}\:\mathrm{and}\:{q}\:\mathrm{are}\:\mathrm{two}\:\mathrm{afirmation} \\ $$$$\sim\left({p}\rightarrow{q}\right)=? \\ $$$$\sim\left({p}\leftarrow{q}\right)=? \\ $$$$\sim\left({p}\Leftrightarrow{q}\right)=? \\ $$

Answered by 112358 last updated on 01/Sep/15

$${We}\:{will}\:{use}\:{the}\:{fact}\:{that}\: \\ $$$${p}\Rightarrow{q}\equiv\backsim{p}\vee{q}.\:{Therefore}, \\ $$$$\backsim\left({p}\Rightarrow{q}\right)\equiv\backsim\left(\backsim{p}\vee{q}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\equiv\backsim\left(\backsim{p}\right)\wedge\backsim{q}\:\:\:\:\:\:{De}\:{Morgan}'{s}\:{law} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\equiv{p}\wedge\backsim{q}\:\:\:\:\:\:\:\:\:\:\:\:.\:\:\:\:\:\:{Involution}\:{law} \\ $$$${Similary},\backsim\left({p}\leftarrow{q}\right)\equiv\backsim\left({q}\Rightarrow{p}\right)\equiv{q}\wedge\backsim{p}\:. \\ $$$${By}\:{definition}, \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:{p}\Leftrightarrow{q}\equiv\left({p}\Rightarrow{q}\right)\wedge\left({q}\Rightarrow{p}\right). \\ $$$$\therefore\backsim\left({p}\Leftrightarrow{q}\right)\equiv\backsim\left[\left({p}\Rightarrow{q}\right)\wedge\left({q}\Rightarrow{p}\right)\right] \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\equiv\backsim\left({p}\Rightarrow{q}\right)\vee\backsim\left({q}\Rightarrow{p}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\equiv\left({p}\wedge\backsim{q}\right)\vee\left({q}\wedge\backsim{p}\right) \\ $$$$ \\ $$$$ \\ $$

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