In each of the following exercises, use associativity, commutativity, and idempotence to simplify the sentence as much as you can using just these rules. your answer should consist of a chain of logical equivalences like the chain given on page 120. at each step of the chain, indicate which principle you are using. here is an example where we use some of these laws to show that the first sentence in the following list is logically equivalent to the last. once again (as in what follows), we use a, b, and c to stand for arbitrary atomic sentences of fol. thus the result is in negation normal form. (a vb) ^ -a) vb) ę (a vb) ^ --a) vb) ☆ (a vb) ^ c^((bva) v b) ☆ (a vb aca(b va vb) (a vb) ^ c^ (b va) ☆ (a vb) ^ c^ (a v b) ☆ (a vb) ac
answer each of the following parts:
a) (b ∧ (a ∧ b ∧ c))
b) (a ∨ b) ∨ (c ∧ d) ∨ a
c) (¬a ∨ b) ∨ (b ∨ c)