Show that p q q is a tautology by developing a series of lo

Show that ¬ (p q) ¬q is a tautology by developing a series of logical equivalences. Please make sure you indicate the name of logical equvaleniceis like implication law, morgan law etc. ( It is the requirement)

Solution

Build a truth table containing each of the statements.

p q ¬q p q ¬(p q) p ¬q

T T F T F F

T F T F TT

F T F TF F

F F T T F F

Since the truth values for ¬(p q) and p¬q are exactly the same for all possible combinations of truth values of p and q, the two propositions are equivalent. Solution 2. We consider how the two propositions could fail to be equivalent. This can happen only if the first is true and the second is false or vice versa. Case 1. Suppose ¬(p q) is true and p ¬q is false. ¬(p q) would be true if p q is false. Now this only occurs if p is true and q is false. However, if p is true and q is false, then p ¬q will be true. Hence this case is not possible. Case 2. Suppose ¬(p q) is false and p ¬q is true. p ¬q is true only if p is true and q is false. But in this case, ¬(p q) will be true. So this case is not possible either. Since it is not possible for the two propositions to have different truth values, they must be equivalent.