Question

Use resolution to show that the hypotheses “It is not raining or Yvette has her umbrella,” “Yvette does not have her umbrella or she does not get wet,” and “It is raining or Yvette does not get wet” imply that “Yvette does not get wet.”

Short answer

Using Resolution on the premises, it can be concluded that Yvette does not get wet.

Step-by-step solution

Proposition and notation

Proposition	Notation
It is raining.	p
Yvette has her Umbrella.	q
Yvette gets wet.	r

Note that \(\neg p\)shows negation of p.

Premises

1. \((\neg p \vee q)\)
2. \((\neg q \vee \neg r)\)
3. \((p \vee \neg r)\)

It is to be proved that \(\neg r\)is true.

Use of Resolution

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

Using Resolution Law on (1) and (3)

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

Using resolution on (2) and above conclusion,

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

\((\neg r \wedge \neg r) \to \neg r\)(by Idempotent law)

Thus \(\neg r\)is true, where r is “Yvette gets wet” then its negation will be “Yvette does not get wet”.