asg1-sol - MATH1090 Problem Set No1 Solutions September...

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MATH1090 Problem Set No1: Solutions September 2007 Faculty of Science and Engineering Dept. of Mathematics and Statistics MATH1090. Problem Set No1: Solutions Posted: Oct. 9, 2007 1. (6 MARKS) (a) Prove that the last symbol of a formula cannot be . (b) Prove that the string ∧∨ cannot appear as a substring in any formula. Your proof in each part will be acceptable only if it is either by induction on formulae , or by analysing formula-calculations . In part (b) you are free to use part (a)’s result even if you did not definitively prove part (a). Part (a) Proof 1. (Analysing Formula-Calculations.) A string is a formula iff it is written at some step of a formula-calculation. There are three kinds of steps: (1) We write a variable, or a constant; neither ends with . (2) We write ( ¬ A ), but do so only if we have already written A . But ( ¬ A ) ends with “)”, not with “ ”. (3) We write ( A B ), where ◦ ∈ {∧ , , , ≡} , but do so only if we have already written the strings A and B . But ( A B ) ends with “)”, not with ”. Proof 2. (Induction on the formula A .) Our claim is: A does not end with . Basis. A is a variable, or a constant. Neither ends with . A is ( ¬ B ). But this ends with “)”, not with “ ”. A is ( B C ). But this ends with “)”, not with “ ”. Noteworthy: As this problem is so easy, the I.H. was neither needed, nor used. Page 1 G. Tourlakis
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MATH1090 Problem Set No1: Solutions September 2007 Part (b) Proof 1. (Analysing Formula-Calculations.) Can a string that con- tains the substring ∧∨ be written at some step of a formula-calculation? Now, the introduction of the “glue”
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This note was uploaded on 05/09/2010 for the course LAPS CSE 1090 taught by Professor Peter during the Winter '10 term at York University.

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asg1-sol - MATH1090 Problem Set No1 Solutions September...

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