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# hw6 - CS 173 Discrete Structures Spring 2010 Homework 6...

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CS 173: Discrete Structures, Spring 2010 Homework 6 This homework contains 5 problems worth a total of 52 regular points. It is due on Friday, March 12th at 4pm. 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 . (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 β is any real number in the range [0 , 1], then ( βx + (1 β ) p, βy + (1 β ) q ) S . iv. If ( x, y ) S , then ( x, y ) S (c) The set M R 2 defined by: i. If x and y are natural numbers and x 2 + y 2

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