[(p q)(q r)](p r)

Developing a chain of logical equivalences which shows the proposition is logically equivalent to T, similar to example 8 of chapter 1.3. Justify each step.

XAMPLE 8 Show that (p ^q) (p vqjis a tautology. Solution: To show that this statement is a tautology, we will use logical equivalences to demon- strate that it is logically equivalent to T. (Note: This could also be done using a truth table.) (p Aq)(p Vq)(pn) V (PVQ) by Example 3 (mp v-y) v (p V q) by the first De Morgan law (op Vp) v q va) by the associative and commutative laws for disjunction by Erample I and the commutative law for disjunction by the domination law