# Thread: Proofs

1. ## Proofs

Prove by direct proof, using only equivalence and inference rules. Remember to name each rule used.
1. (p ^ q) → r

2. p ^ ¬ r

Therefore ¬q.

2. $\begin{gathered}
\left( {p \wedge q} \right) \to r \hfill \\
p \wedge \neg r \hfill \\
\------- \hfill \\
p \hfill \\
\neg r \hfill \\
\neg \left( {p \wedge q} \right) \hfill \\
\neg p \vee \neg q \hfill \\
\neg q \hfill \\
\end{gathered}$

You fill in the reasons.

3. Hello, captainjapan!

$(p \to q) \;\Leftrightarrow\;\sim p \vee q$ . Alternate definition of implication (ADI)

Prove by direct proof, using only equivalence and inference rules.
Remember to name each rule used.

$\begin{array}{c}(p \wedge q) \to r \\ p \:\wedge \sim\! r \\ \hline
\therefore\;\sim\! q \end{array}$

. . $\begin{array}{ccc}
(p \wedge q ) \to r & & \text{Given} \\
\sim(p \wedge q) \vee r & & \text{ADI} \\
(\sim\!p \:\vee \sim\!q) \vee r & & \text{DeMorgan} \\
(\sim\! p \vee r) \:\vee \sim\! q & & \text{comm/assoc} \\
\sim(\sim\!p \vee r) \to \:\sim\!q & & \text{ADI} \\
(p \:\wedge \sim\!r) \to \:\sim\! q & & \text{DeMorgan} \\
p \:\wedge \sim r & & \text{Given} \\
\therefore\;\;\sim\!q & & \text{Modus Ponens}
\end{array}$

4. Originally Posted by Soroban
$(p \to q) \;\Leftrightarrow\;\sim p \vee q$ . Alternate definition of implication (ADI)
In almost all standard logic textbooks the property
$(p \to q) \;\Leftrightarrow\;\sim p \vee q$ is known as Material Implication (Impl.).