hw6sol - CS 173: Discrete Structures, Spring 2010 Homework...

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Unformatted text preview: CS 173: Discrete Structures, Spring 2010 Homework 6 Solutions This homework contains 5 problems worth a total of 52 regular points. Since the main point of this assignment is to learn how to write proofs by induction, you must use this proof technique when the problem says to use it, even if a non-inductive proof is also possible. 1. Recursive definition [12 points] Give a simple closed-form definition for each of the following recursively-defined sets. Give both a precise definition using set-builder notation, also an informal description using a picture and/or words, and an informal explanation or work. (a) The set T Z 2 defined by: i. (1 , 1) T ii. (2 , 2) T iii. If ( x,y ) T , then ( x + 1 ,y ( x + 1)) T . Solution (4pts): T = { ( n,n !) | n 1 } The set T consists of all pairs ( n,n !) where n 1 and n ! = n ( n 1) 2 1. This can be proved by induction, given the facts that the formula holds for the points (1 , 1); (2 , 2) and given a point ( x,x !), rule three constructs the point ( x + 1 , ( x + 1) x !) = ( x + 1 , ( x + 1)!), for which the formula still holds. (b) The set S R 2 defined by: i. (3 , 4) S ii. ( 2 , 4) S iii. If ( x,y ) S and ( p,q ) S , and is any real number in the range [0 , 1], then ( x + (1 ) p, y + (1 ) q ) S . iv. If ( x,y ) S , then ( x, y ) S Solution (4pts): S = { ( x,y ) | 2 x 3 , 4 y 4 } The set S consists of all points in the closed, axis-aligned rectangle with sides at x = 2 , 3 and y = 4 , 4. (c) The set M R 2 defined by: i. If x and y are natural numbers and x 2 + y 2 = 25, then ( x,y ) M , ii. If ( x,y ) M and k is a positive integer, then ( kx,ky ) M ....
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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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hw6sol - CS 173: Discrete Structures, Spring 2010 Homework...

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