# Homework 4

## Table of Contents

## Task (100 pts)

- (10pts) Build truth tables for the following formulas:
- \((A \vee B) \wedge C\)
- \((A \rightarrow B) \vee (B \rightarrow \neg C)\)

- (10pts) Find variable assignments that satisfy the following
formulas, or state that no such assignment exists:
- \((A \wedge \neg B) \vee (\neg A \wedge B \wedge C)\)
- \(C \rightarrow (\neg C \wedge \neg A)\)

- (10pts) Express the following statements in propositional logic:
- \(B\) is true whenever \(C\) is false.
- Either \(X\) is true or \(Z\) is true but not both.
- At most one of \(R\), \(S\), and \(T\) can be true.
- The dog only barks when she hears a noise. (Create propositional symbols, and give their definitions.)

- (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:

- \(Z\)
- \(\neg P\)
- \(\neg (J \wedge Q)\)
- \(\neg J\)
- \(H\)

- (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…)- James is a dolphin.
- Dolphins are mammals.
- Mammals have hair and three middle ear bones.
- Some mammals have sweat glands and specialized teeth.
- Vignesh stopped by his house before going to the party.
- There is no apparent correlation between tobacco use and city of birth.
- The result of firing a loaded gun at a person is that the person dies.
- 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…) - Obama will win the 2012 election.
- There is a chance that Obama will win the 2012 election.
- I’m hungry.
- This sentence is false.

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