hw6-sol - CS 173: Discrete Structures, Fall 2010 Homework 6...

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Unformatted text preview: CS 173: Discrete Structures, Fall 2010 Homework 6 Solutions This homework was worth a total of 54 points. 1. Recursive definition [13 points] Give a simple closed-form definition for each of the following subsets of the real plane. Give both a precise definition using set-builder notation and also an informal geometrical description using a picture and/or words. (a) (4 points) The set T defined by: i. (1 , 1) T , ii. If ( x,y ) T , then ( x + 1 ,y + 1) T , iii. If ( x,y ) T , then ( x + 2 ,y ) T , iv. If ( x,y ) T , then ( x,y + 2) T . [Answer:] T = { ( x,y ) ( Z + ) 2 : x,y Z + and x+y is even } . That is, the set T consists of all ordered pairs of positive integers in the plane where the sum of the elements in each pair is even. (b) (4 points) The set T R 2 defined by: i. (0 , 0) T , ii. If ( x,y ) T , then ( x + r, ( x + r ) 2 ) T where r is some real number 0, iii. If ( x,y ) T , then (- x,y ) T . [Answer:] T = { ( x,y ) R 2 : y = x 2 } . That is theset T contains all points on the parabola y = x 2 . The next two problems are related. (c) (2 points) The set T R 2 containg all points of the form (3 , p 9- (3 ) 2 ) T , is a real number in the range [0 , 1], (Assume that x returns the positive square root of x .) [Answer:] T = { ( x,y ) : x,y 0 and x 2 + y 2 = 9 } . That is set T contains all points ( x,y ) in the first quadrant which are on the circle of radius 3 centered at the origin. 1 (d) (3 points) The set T R 2 defined by: i. For all real numbers in the range [0 , 1], (3 , p 9- (3 ) 2 ) T , ii. If ( x,y ) T , then ( x,- y ) T , iii. If ( x,y ) T , then (- x,y ) T . iv. If ( x,y ) T and ( p,q ) T then for some real number in the range [0 , 1], ( x,y ) + (1- )( p,q ) T . [Answer:] T = { ( x,y ) R 2 : x 2 + y 2 9 } . That is, the set T contains all points on and inside the circle of radius 3 centered at the origin....
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hw6-sol - CS 173: Discrete Structures, Fall 2010 Homework 6...

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