For each of the following pairs of propositions, show that the two propositions are logically equivalent by finding a chain of equivalences from one to the other. State which definition or law of logic justifies each equivalence in the chain.

a$)p^{{?}^{?}}(q^{{?}^{?}}p),p^{{?}^{?}}q$

b$)(notp)q,pvvq$

c$)(pvvq{)}^{{?}^{?}}$notq,$p^{{?}^{?}}$notq

d$)p(qr),(p^{{?}^{?}}q)r$

e$)(pr{)}^{{?}^{?}}(qr),(pvvq)r$

f$)p(p^{{?}^{?}}q),pq$

For each of the following compound propositions, find a simpler proposition that is logically equivalent. Try to find a proposition that is as simple as possible.

a$)(p^{{?}^{?}}q)vvnotq$

b$)$ not$(pvvq{)}^{{?}^{?}}p$

c$)p$notp

d$)not{p}^{{?}^{?}}(pvvq)$

e$)(q^{{?}^{?}}p)q$

f$)(pq{)}^{{?}^{?}}(notpq)$