Chapter 1.3, Problem 13E

Show that each conditional statement in Exercise 11 is a tautology using the fact that a conditional statement false exactly when the hypothesis true and the conclusion is false. (Do not use truth tables.)

11. Show that each of these conditional statements is a tautology by using truth tables.

a) $\left(p\wedge q\right)\to p$

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

c) $\neg p\to \left(p\to q\right)$

d) $\left(p\wedge q\right)\to \left(p\to q\right)$

e) $\neg \left(p\to q\right)\to p$

f)

   $\neg \left(p\to q\right)\to \neg q$