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# Homework 4

1. (10pts) Build truth tables for the following formulas:
1. $$(A \vee B) \wedge C$$
2. $$(A \rightarrow B) \vee (B \rightarrow \neg C)$$
2. (10pts) Find variable assignments that satisfy the following formulas, or state that no such assignment exists:
1. $$(A \wedge \neg B) \vee (\neg A \wedge B \wedge C)$$
2. $$C \rightarrow (\neg C \wedge \neg A)$$
3. (10pts) Express the following statements in propositional logic:
1. $$B$$ is true whenever $$C$$ is false.
2. Either $$X$$ is true or $$Z$$ is true but not both.
3. At most one of $$R$$, $$S$$, and $$T$$ can be true.
4. The dog only barks when she hears a noise. (Create propositional symbols, and give their definitions.)
4. (20pts) We have a knowledge base that contains:
• $$(R \rightarrow S) \wedge Q$$
• $$(J \wedge Q) \rightarrow P$$
• $$Q \rightarrow Z$$
• $$P \rightarrow \neg Q$$
• $$(R \rightarrow S) \rightarrow (\neg H \rightarrow J)$$

Derive each of the following using inference rules for propositional logic:

1. $$Z$$
2. $$\neg P$$
3. $$\neg (J \wedge Q)$$
4. $$\neg J$$
5. $$H$$
5. (40pts) Rewrite the following statements in first-order logic, stating definitions for the predicates you use. (Note, some of these are hard; make an attempt…)
1. James is a dolphin.
2. Dolphins are mammals.
3. Mammals have hair and three middle ear bones.
4. Some mammals have sweat glands and specialized teeth.
5. Vignesh stopped by his house before going to the party.
6. There is no apparent correlation between tobacco use and city of birth.
7. The result of firing a loaded gun at a person is that the person dies.
8. The usual result of firing a loaded gun at a person is that the person dies (unless, the person has a bulletproof vest, or the gun misfires, or…)
9. Obama will win the 2012 election.
10. There is a chance that Obama will win the 2012 election.
11. I’m hungry.
12. This sentence is false.