Chapter 1.3, Problem 3E

Verify that each of the following arguments is valid.

(a)

$\begin{array}{l}p\to r\\ q\to r\end{array}$

$\left(p\vee q\right)\to r$

(b)

$\begin{array}{l}p\to r\\ q\to s\end{array}$

$\left(p\wedge q\right)\to \left(r\wedge s\right)$

(c)

$\begin{array}{l}p\vee q\\ \left(\neg p\right)\vee r\\ \left(\neg r\right)\vee s\end{array}$

$q\vee s$

(d)

$\begin{array}{l}p\vee \left(\left(\neg q\right)\wedge r\right)\\ \neg \left(p\wedge s\right)\end{array}$

$\neg \left(s\wedge \left(q\vee \left(\neg r\right)\right)\right)$