Please write so I can copy paste it****

Exercise 1.5.1: Label the steps in a proof of logical equivalence.

Below are several proofs showing that two logical expressions are logically equivalent. Label the steps in each proof with the law used to obtain each proposition from the previous proposition. The first line in the proof does not have a label.

(a)

(p q) (q p)

(p q) (q p)

(q p) (q p)

q (p p)

q (p p)

q F

q

(b)

(p q) (p q)

(p q) (p q)

(p q) (p q)

(p q) (p q)

p (q q)

p T

p

(c)

r (r p)

r (r p)

r (r p)

(r r) p

r p