1. (30 points) Prove the following statement is a tautology without using truth tables use the logical equivalences from Table G-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)-(q V r) -p (b) (8 points) p (q Vr) (pA-q) r (c) (8 points) [-pn(pv q)]-+ q T (d) ( 10 points) [(p V q) (p r) (q-n]-t r T 2. (8 points) Show the following expressions are logically equivalent. If you are not lazy, then you work hard or you are clever and lazy You work hard or you are lazy Use the following propositions to represent the concepts in the expression. Let . be "you are lazy" w be "you work hard . c be you are clever 3, (8 points) Rosen Ch 1.6 #6, p. 78 Let r be the proposition "It rains", let f be "It is foggy", let s be "The sailing race will be held", let be "The life saving demonstration will go on", and let t be "The trophy will be awarded