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# ex3 - CS70 Discrete Mathematics and Probability Spring 2012...

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CS70 Discrete Mathematics and Probability, Spring 2012 Homework 3 Out: 2 Feb. Due: 5pm, 9 Feb. Instructions: Start each problem on a new sheet. Write your name, section number and “CS70” on every sheet. If you use more than one sheet for a problem, staple them together (but do not staple different problems together). Put your solutions in the boxes on Soda level 2 by 5pm on Thursday: your solution to Q1 goes in box CS70–1, your solution to Q2 goes in box CS70–2, and so on ( ﬁve boxes total). You are encouraged to form small groups (two to four people) to work through the homework, but you must write up all your solutions on your own. 1. [Strengthening the Induction Hypothesis] Prove that, for all n 1 , all entries of the matrix ± 1 0 1 1 ² n are bounded above by n . [HINT: Look at the title of this problem! To come up with a stronger hypothesis, try evaluating the matrix for the ﬁrst few values of n .] 2. [ Well-Ordering Principle] This problem concerns the Well-Ordering Principle. First, we will see that the set of all pairs of natural numbers N × N = { ( a,b ) : a N ,b N } is a well-ordered set, provided we deﬁne the ordering correctly. Then we will use this fact to do an inductive proof over pairs of natural numbers. (a) Suppose we deﬁne the ordering on N × N by ( a,b ) ( c,d ) if and only if either a < c , or a = c and b < d . (This is usually called the “lexicographic ordering.”) Show, using the Well-Ordering Principle for the natural numbers N , that N × N is well-ordered with this ordering.

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ex3 - CS70 Discrete Mathematics and Probability Spring 2012...

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