Section 1.2

• 16. Both expressions are F when p is T and q is F, and both expressions are T for the other three cases, so they are logically equivalent. (This is just a textual description of the truth tables you could construct for this.)
• 18. Both are T when p and q have the same truth value, and both are F with p and q have different truth values, so they are logically equivalent.
• 20. a) p V -q V -r; b) (p V q V r) ^ s; c) (p ^ T) V (q ^ F)
• 26. For each row in the truth table for which the expression is true, express the row as a conjunction of the truth values of each simple proposition for that row. Combine all of these conjunctions with OR to make one big disjunction.
• 28. By exercise 27, we can write any proposition using just V, ^, and ~ (not). By DeMorgan's Law we can get rid of all V's by replacing each occurrence of p1 V p2 V ... V pn with ~ (~p1 ^ ~p2 ^ ... ^ ~pn).
• 34. a) From truth tables for ~p and for p NOR q we see they are logically equivalent. b) From part a, (p NOR q) NOR (p NOR q) is same as ~(p NOR q), which, by definition or truth table, we know is equivalent to p V q. c) By exercise 29, ~ and V are functionally complete. Thus, NOR's can be converted into ~'s and V's, so NOR's are functionally complete.
• 40. This can be argued in terms of a truth table.