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# ps1 - Massachusetts Institute of Technology 6.042J/18.062J...

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Massachusetts Institute of Technology 6.042J/18.062J, Spring ’11 : Mathematics for Computer Science February 2 Prof. Albert R Meyer revised Sunday 6 th February, 2011, 03:18 Problem Set 1 Due : February 11 Reading: Part I . Proofs: Introduction , Chapter 1 , What is a Proof? ; Chapter 2 , The Well Ordering Principle ; and Chapter 3 through 3.5 , covering Propositional Logic . These assigned readings do not include the Problem sections . (Many of the problems in the text will appear as class or homework problems.) Reminder : Email comments on the reading are due at times indicated in the online tutor problem set TP.2. Reading Comments count for 3% of the final grade. Problem 1. The fact that that there are irrational numbers a; b such that a b is rational was proved in Problem 1.2 of the course text. Unfortunately, that proof was nonconstructive : it didn’t reveal a specific pair, a; b , with this property. But in fact, it’s easy to do this: let a WWD p 2 and b WWD 2 log 2 3 . We know p 2 is irrational, and obviously a b D 3 . Finish the proof that this a; b pair works, by showing that 2 log 2 3 is irrational. Problem 2. Use the Well Ordering Principle to prove that n 3 n=3 (1) for every nonnegative integer, n . Hint: Verify ( 1 ) for n 4 by explicit calculation. Problem 3. Describe a simple recursive procedure which, given a positive integer argument, n , produces a truth table whose rows are all the assignments of truth values to n propositional variables. For example, for n D 2 , the

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