# hw5-soln - CS 173: Discrete Structures, Spring 2010...

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Unformatted text preview: CS 173: Discrete Structures, Spring 2010 Homework 5 Solutions This homework contains 5 problems worth a total of 50 regular points, plus three bonus points. It is due on Friday, March 5th at 4pm. 1. Functions [8 points] For each of the following functions, give the following information: what is its co- domain? what is its image? is the function onto? is the function one-to-one? (a) f : Z Z such that f ( x ) = 2 b x 2 c Solution: Codomain : Z Image : { n Z : n = 2 m for some m Z } , i.e., the even integers One-to-one ? No. For instance, f (0) = 0 = f (1). Onto ? No. The codomain is Z , but no odd integers are in the image of f. (b) g : N N such that g ( x ) = x ( x +1) 2 Solution: Codomain : N Image : { , 1 , 3 , 6 , 10 , 15 , 21 , . . . } Notice that the difference between sequential numbers in this list increases by one as you read from left to right. One-to-one ? Yes. Notice that the sequence is strictly increasing. Onto ? No. For instance, 2 is not in the image of g . (c) h : C R such that h ( ai + b ) = a (where i = - 1) Solution: Codomain : R Image : R One-to-one ? No. For instance, h (2 i + 1) = 2 = h (2 i + 2). Onto ? Yes, since the image and codomain are equal. (d) k : R 2- { (0 , 0) } R 2 such that k ( x, y ) = ( x l ( x,y ) , y l ( x,y ) ) where l ( x, y ) = p x 2 + y 2 Solution: Codomain : R 2 Image : The unit circle, S 1 = { ( a, b ) R 2 : a 2 + b 2 = 1 } ....
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## This note was uploaded on 09/21/2011 for the course CS 173 taught by Professor Erickson during the Spring '08 term at University of Illinois, Urbana Champaign.

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hw5-soln - CS 173: Discrete Structures, Spring 2010...

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