1. The equivalences follow by showing that the appropriate pairs of columns of the following table agree.

 p p F p T p p T F T T F F T F

2.

 p     q     r q r p (q r) p q p r (p q) (p r) T    T    T T T T T T T    T    F T T T F T T    F    T T T F T T T    F    F F F F F F F    T    T T F F F F F    T    F T F F F F F    F    T T F F F F F    F    F F F F F F

3. a) If the hypothesis p is true, by the definition of disjunction, the conclusion p q is also true.

If p is false on the other hand, then by the definition of implication p (p q)

is true.

Altenatively, p (p q) ( p (p q)) (( p p ) q) (T q) T

b) If the hypothesis p q is true, then both p and q are true so that the conclusion p q is also true. If the hypothesis is false, then "if-then" statement is always true.

This can also be proven similarly to the alternative proof for a).

c) If the hypothesis (p q) is true, then p q is false, so that p is true and q is false.  Hence, the conclusion q is true.If the hypothesis is false, then "if-then" statement is always true.

This can also be proven similarly to the alternative proof for a).

4. a) If p is true, then p (p q) is true since the first proposition in the disjunction is true.  On the other hand, if pis false, then p q is also false, so p (p q) is false.  Since p and p (p q) always have the same truth value, they are equivalent.

This can also be proven similarly to b).

b) [ p (p q) ]

[ (p F ) (p q) ]

[ (p ( F q) ] [ p F ] p

This can also be proven similarly to a).

5. a) (p q r)

b) (p q r) s

c) (p T) (q F)

6. (p q r ) ( p q r ) ( p q r )