hw5 - 1. If h is one-to-one, then either f or g must be...

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CSE 20: Discrete Mathematics Fall 2010 Problem Set 5 Instructor: Daniele Micciancio Due on: Wed Nov 10, 2010 In addition to the problems below, you should also solve the following problems from the textbook for additional training, but without submitting your answers: Section 2.2 (8,10), Section 2.3 (8, 9,10,13,16,23,25), Section 2.4 (7,8,12,24,25). Problem 1 (8 points) For each of the following functions says if the function is injective (1-to-1) and/or surjective (onto). 1. The function f : Z × ( Z \ { 0 } ) Q defined by the rule f ( a,b ) = a/b . 2. The function f : Z Z × Z defined by the rule f ( x ) = (2 x + 3 ,x - 4) 3. The function f : Z × Z Z × Z defined by the rule f ( x,y ) = (2 x + y, 2 x - y ) 4. The function f : Q × Q Q × Q defined by the rule f ( x,y ) = (2 x + y, 2 x - y ) Problem 2 (8 points) Let f : X Y , g : Y Z and defined the function composition h = g f : X Z by h ( x ) = g ( f ( x )) . Prove or disprove each of the following assertions:
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Unformatted text preview: 1. If h is one-to-one, then either f or g must be one-to-one 2. If h is a bijection, then both f and g must be surjective Problem 3 (9 points) For each of the following relations, determine if it is an equivalence relation. If so, list its equivalence classes. If not, says which properties (reexive, symmetric and transitive) it fails to satisfy. (If it fails more than one property, list all of them.) 1. The relation R on { 1 , 2 , 3 , 4 , 5 , 6 } dened by aRb 2 | ( a + b ) , i.e., ( a,b ) is in the relation if a + b is even 2. The relation R on the powerset 2 { 1 , 2 , 3 } where ( A,B ) R if and only if A B = 3. The relation R on the set { 1 , 2 , 3 , 4 , 5 , 6 } dened by aRb 3 | ( a + b )...
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