The Associative Laws (see Table 6 in section 1.3.2) justify writing expressions involving only or only without parentheses. Since (pq)r p(q r), we can safely write pq r to represent both, without worrying about how someone might interpret it. We can also use the associative laws to show that all of parenthesized expressions involving four atomic propostions are logically equivalent. Example If we define t to be the compound proposition p q, then we can see that (p q) r s (t r) s by substitution of t p q t (r s) by the associative law for (p q) (r s) by substitution of p q t Note: We may not always make the t substitutions explicit. With practice, we will become comfortable summarizing the above derivation as: (p q) r s (p q) (r s) by the associative law applied to p q, r, and s.

(a) Find a similar sequence of equivalences that proves p (q r) s p q (r s) .

(b) The expression p q r s can be parenthesized in 5 different ways. Four of these ways are listed in either the example above or part (a), what is the fifth way to parenthesize p q r s?

(c) Use logical equivalences to show that your expression in (b) is equivalent to one of the expressions in (a).

(d) Use logical equivalences to show that your expression in (b) is equivalent to one of the expressions in the example.

Please answer all the 4 parts of the question.