1.) a,b,c,d is what I have questions about. If you need help with the format, their is the example from the book to help.

Thanks!

1. (30 points) Prove the following statement is a tautology without using truth tables (use the logical equivalences from Table 6-8 of the book). Justify each step with the law used. Model the solutions in the style of Examples 6-8, pp. 29-30 of the book. (a) (4 points) p (-q- r)--4 Vr)-+-p (b) (8 points) p (q V r) (pA-q) r (c) (8 points) pA (pV )qT (d) (10 a) A (p)A( T points) (p V EXAMPLE 7 Show that-p V ( q)) and p ^-y are logically equivalent by developing a series of logical equivalences Solution We will use one of the equivalences in Table 6 at a time, starting with (P ( ^ q)) and ending with ^-'q. (Note: we could also easily establish this equivalence using a truth table.) We have the following equivalences. by the second De Morgan law by the first De Morgan law by the double negation law by the second distributive law because P F by the commutative law for disjunction by the identity law for F ^^ [(mp) v-v] 7p p) V ( -v) Consequently (p V ( q)) and are logically equivalent EXAMPLE 8 Show that (p q) (p V q) Is 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.) (1p v-y ) V (p V q) (1p v p) v (-q V q) by the first De Morgan law by the associative and commutative by Example 1 and the commutative by the domination law laws for disjunction law for disjunction