Solutions17 - Chapter 17 Hints and Selected Solutions...

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Chapter 17: Hints and Selected Solutions Section 17.1 (page 470) 17.1 Here for your convenience is the truth table ( P , Q , R ): P Q R ( P , Q , R ) t t t T t t f T t f t F t f f F f t t T f t f F f f t T f f f F This can be nicely captured as follows: ˆ h ( ( P , Q , R )) = ˆ h ( Q ) if ˆ h ( P ) = true ; ˆ h ( ( P , Q , R )) = ˆ h ( R ) if ˆ h ( P ) = false There are other ways of expressing the same thing, though. Just make sure that your definition gives the truth table above. 17.3 Assumethat h 1 and h 2 are truth assignments that assign the same value to the atomic sentences in S . We are asked to prove that ˆ h 1 ( S ) = ˆ h 2 ( S ). We prove this by induction on wffs. Basis: In this case S is itself atomic, so the assumption just immediately gives the result. Induction Step . There are several cases to consider, corresponding to the ways of building up propositional wffs. Here is one of the cases. Suppose that we know the result for P and Q and want to show that it is true for P Q . Our induction hypothesis insures us that ˆ h 1 ( P ) = ˆ h 2 ( P ) and ˆ h 1 ( Q ) = ˆ h 2 ( Q ). We know that ˆ h 1 ( P Q ) = true if and only if ˆ h 1 ( P ) = true or ˆ h 1 ( Q ) = true , or both, by the definition of ˆ h 1 (i.e.
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