Chapter 1.3, Problem 14E

Show that each conditional statement in Exercise 12 is a tautology using the fact that a conditional statement is false exactly when the hypothesis is true and the conclusion is false. (Do not use truth tables.)

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

a) $\left[\neg p\wedge \left(p\vee q\right)\right]\to q$

b) $\left[\left(p\to q\right)\wedge \left(q\to r\right)\right]\to \left(p\to r\right)$

c) $\left[p\wedge \left(p\to q\right)\right]\to q$

d)

   $\left[\left(p\vee q\right)\wedge \left(p\to r\right)\wedge \left(q\to r\right)\right]\to r$