Question

Use resolution to show the hypotheses “Allen is a bad boy or Hillary is a good girl” and “Allen is a good boy or David is happy” imply the conclusion “Hillary is a good girl or David is happy.”

Short answer

Using Resolution, it can be concluded that either Hillary is a good girl or David is Happy.

Step-by-step solution

Assign notation

Let pbe the proposition that “Allen is a good boy.”

Then its negation \(\neg p\)will be “Allen is a bad boy.”

Let q be the proposition that “Hillary is a good girl.”

Let r be the proposition that “David is happy.”

Premises

Allen is a bad boy or Hillary is a good girl.	\((\neg p \vee q)\)
Allen is a good boy or David is happy.	\((p \vee r)\)

Resolution

Resolution Law:\(((p \vee q) \wedge (\neg p \vee r)) \to (q \vee r)\)where p, q and r are propositions.

Using Resolution Law in the given premises

\(((\neg p \vee q) \wedge (p \vee r)) \to (q \vee r)\)

Hence\((p \vee r)\)is true.

So, either Hillary is a good girl or David is happy.