Answer to Question #306327 in Discrete Mathematics for paul

Question

1. –p → q
2.  (p → –q) ∨ –q
3.  p → –(p ∨ q)
4. (p ∧ q) ∨ (–p ∨ q)
5. [ (p ∧ q) ∨ –p ] ∧ –q

Answer 1

1)

$$\begin{matrix} \hline p & q & -p & -p\rightarrow q \\ \hline 0 & 0 & 1 & 0 \\ \hline 0 & 1 & 1 & 1 \\ \hline 1 & 0 & 0 & 1 \\ \hline 1 & 1 & 0 & 1 \\ \hline \end{matrix}$$

2)

$$\begin{matrix} \hline p & q & -q & p\rightarrow -q & (p\rightarrow-q)\vee-q \\ \hline 0 & 0 & 1 & 1 & 1 \\ \hline 0 & 1 & 0 & 1 & 1 \\ \hline 1 & 0 & 1 & 1 & 1 \\ \hline 1 & 1 & 0 & 0 & 0 \\ \hline \end{matrix}$$

3)

$$\begin{matrix}\hline p & q & p\vee q & -(p\vee q) & p\rightarrow -(p \vee q) \\ \hline 0 & 0 & 0 & 1 & 1 \\ \hline 0 & 1 & 1 & 0 & 1 \\ \hline 1 & 0 & 1 & 0 & 0 \\ \hline 1 & 1 & 1 & 0 & 0 \\ \hline \end{matrix}$$

4)

$$\begin{matrix}\hline p & q & p \land q & -p & -p \vee q & (p \land q)\vee(-p \vee q)) \\ \hline 0 & 0 & 0 & 1 & 1 & 1 \\ \hline 0 & 1 & 0 & 1 & 1 & 1 \\ \hline 1 & 0 & 0 & 0 & 0 & 0 \\ \hline 1 & 1 & 1 & 0 & 1 & 1 \\ \hline \end{matrix}$$

5)

$$\begin{matrix}\hline p & q & p \land q & -p & (p \land q)\vee-p & -q & {[}(p \land q) \vee -p{]} \land -q \\ \hline 0 & 0 & 0 & 1 & 1 & 1 & 1 \\ \hline 0 & 1 & 0 & 1 & 1 & 0 & 0 \\ \hline 1 & 0 & 0 & 0 & 0 & 1 & 0 \\ \hline 1 & 1 & 1 & 0 & 1 & 0 & 0 \\ \hline \end{matrix}$$