Math+150++F09-+Hwk+3+Solutions

# Math+150++F09-+Hwk+3+Solutions - SOLUTIONS TO HOMEWORK 3...

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Unformatted text preview: SOLUTIONS TO HOMEWORK 3 MATH 150, FALL 09 Problem 1. Section 1.5/ Exercise 12 {âˆ§ , > , âŠ¥} is not complete. To show that, we claim that for every Î± with one sentence symbol A , and connective symbols among âˆ§ , > , âŠ¥ , either Î± is a tautology, a contradiction, or A | = Î± . The proof, as usual, is by induction on Î± : (1) If Î± is A , or > , or âŠ¥ , then the conclusion is clear. (2) If Î± = Î² âˆ§> , then Î± is equivalent to Î² . Using the induction hypoth- esis for Î² , we get the conclusion for Î± . (3) If Î± = Î² âˆ§ âŠ¥ , then Î± is equivalent to âŠ¥ , and so Î± is a contradiction. (4) If Î± = Î² âˆ§ Î³ , then we have a the following cases: â€¢ if both Î² and Î³ are tautologies, then so is Î± . â€¢ if at least one of Î² and Î³ is a contradiction, then so is Î± . â€¢ if A | = Î² and either A | = Î³ or Î³ is a tautology, then A | = Î± . â€¢ if A | = Î³ and either A | = Î² or Î² is a tautology, then A | = Î± ....
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## This note was uploaded on 11/16/2009 for the course MATH 120 taught by Professor Smith during the Spring '09 term at UC Irvine.

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Math+150++F09-+Hwk+3+Solutions - SOLUTIONS TO HOMEWORK 3...

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