Answer:
(a) p v ~q
(b) ~(p v q) v ~p
(c) ~(~p v ~q) v (p v q)
Step-by-step explanation:
The conditional-disjunction equivalence is:
P→Q ⇔ ~P v Q
To find an equivalent compound proposition without the conditionals (without the "→") you have to apply the previous equivalence and simplify if possible.
a) ~p→~q
In this case, P= ~p and Q= ~q
Applying the equivalence:
~(~p) v ~q
p v ~q
b) (p v q) → ~p
In this case P = (p v q) and Q= (~p)
Applying the equivalence:
~(p v q) v ~p
c) (p→~q) → (~p→q)
In this case, you have to apply the conditional-disjunction equivalence for every conditional in the compound proposition.
First, let P= (p→~q) and Q= (~p→q)
~ (p→~q) v (~p→q) (1)
Now, you have to find an equivalent compound proposition for both (p→~q) and (~p→q)
For (p→~q):
Let P= p and Q=~q
~p v ~q
For (~p→q)
Let P= ~p and Q= q
~(~p) v q
p v q
Then the expression (1) is:
~(~p v ~q) v (p v q)