16.410/16.413 Recitation 7 Solutions
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Solution: Let us define the atomic propositions, Julia, Sarah, and Fred to indicate that Julie is telling the truth, Sarah is telling the truth, and Fred is telling the truth, respectively. First notice that if A lies then the opposite of A’s statement is true, i.e., if A claims that B and C tell the truth but A lies, then we have “(not A) implies (not(B and C))”, which is equivalent to “(not A) implies ((not B) or (not C))” by de Morgan’s law. Now we can write the following several statements, each of which is true for the given English sentence. A1. Julia implies (Sarah and Fred) A2. (not Julie) implies ((not Sarah) or (not Fred)) A3. Sarah implies (not Peter) A4. (not Sarah) implies Peter A5. Peter implies Fred A6. (not Peter) implies (not Fred) A7. Fred implies ((not Peter) or (not Julia)) A8. (not Fred) implies (Peter and Julia) A9. Dick implies (not Dick) A10. (not Dick) implies Dick The sentence is basically a conjunction of the all the statements above, i.e., (Julia implies (Sarah and Fred)) and ((not Julie) implies ((not Sarah) or (not Fred))) and (Sarah implies (not Peter)) and ((not Sarah) implies Peter) and (Peter implies Fred) and ((not Peter) implies (not Fred)) and (Fred implies ((not Peter) or (not Julia))) and ((not Fred) implies (Peter and Julia)) (Dick implies (not Dick)) and ((not Dick) implies Dick)
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