Assignment2

Assignment2 - Math 240, Fall 2009 — Assignment 2 — Due:...

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Unformatted text preview: Math 240, Fall 2009 — Assignment 2 — Due: Monday, October 5 1. Suppose that five ones and four zeros are arranged around a circle. Let’s use the word bit to mean either a zero or a one. We do the following steps: Step 1: Between any two equal bits, insert a 0 and between any two unequal bits, insert a 1. This produces nine new bits. Step 2: Erase the nine original bits. Prove that when we perform this procedure repeatedly, we never have nine zeros around the circle after Step 2. Hint: Give a proof by contradiction. Work backward, assuming that you did end up with nine zeros. 2. Suppose that A, B , C are sets. Prove that (B − A) ∪ (C − A) = (B ∪ C ) − A. 3. Suppose A and B are two finite sets. Recall that P (A) is the power set of A, which is the set of all subsets of A. (a) Prove, or disprove, that P (A ∩ B ) = P (A) ∩ P (B ). (b) Prove, or disprove, that P (A ∪ B ) = P (A) ∪ P (B ). 4. Each of the following functions f is a function from R to R. Determine whether each f is injective (one-to-one), surjective (onto), or both. Please give reasons. (a) f (x) = −3x + 4 (b) f (x) = −3x2 + 7 (c) f (x) = x+1 x+2 1 if x = −2 if x = −2 (d) f (x) = x5 − 1 (e) f (x) = 2 x − x 5. (Peer reviewed question) Prove that if S is a set, there does not exist a surjective function from S to its power set P (S ). What does this tell you about the cardinalities of infinite sets? Discuss. Hint: Suppose that we have a function f : S → P (S ). Let T = {s ∈ S |s ∈ f (s)} ∈ P (S ) Show that T has no preimage under f — and conclude that f isn’t surjective. 6. Give a big-O estimate for each of these functions, of the form “f (x) is O(g (x))”. For the function g in your estimate, use a simple function g of smallest possible order. (a) f (x) = (n3 + n2 log n)(log n + 1) + (17 log n + 19)(n3 + 2) (b) f (x) = (2n + n2 )(n3 + 3n ) (c) f (x) = (nn + n · 2n + 5n )(n! + 5n ) Math 240 — Assignment 2 — Fall 2009 Marker: Jonathan Cottrell Instructor: Benjamin Young Date: Name: Student Number: ...
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This note was uploaded on 10/03/2010 for the course MATH 240 taught by Professor Szabo during the Spring '08 term at McGill.

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Assignment2 - Math 240, Fall 2009 — Assignment 2 — Due:...

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