quiz3-review-solutions

# quiz3-review-solutions - x 0 If x,y =(0 4 then the equation...

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CS 173: Discrete Structures, Spring 2010 Quiz 3 review solutions 1. Relation properties A B C D Refexive : Irrefexive : Symmetric : Antisymmetric : Transitive : is the relation on R such that x y if and only if xy = 1 Refexive : Irrefexive : Symmetric : Antisymmetric : Transitive : A B C D E Refexive : Irrefexive : Symmetric : Antisymmetric : Transitive : 1

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2. Equivalence classes Let A = R + × R + -{ (0 , 0) } , i.e. pairs of positive reals in which no more than one of the two numbers is zero. Consider the equivalence relation on A deFned by ( x,y ) ( p,q ) ( xy )( p + q ) = ( pq )( x + y ) (a) List four elements of [(3 , 1)]. Hint: what equation do you get if you set ( x,y ) to (3 , 1) and q = 2 p ? (b) Give two other distinct equivalence classes that are not equal to [(3 , 1)]. (c) Describe the members of [(0 , 4)]. Solutions: (a) (3 , 1), (1 , 3), ( 9 8 , 9 4 ), ( 9 4 , 9 8 ). You can Fnd a range of other elements by setting q to other multiples of p . (b) ²or example, [(3 , 2)], [(3 , 4)] (c) All pairs of the form (0 ,y ) or (
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Unformatted text preview: x, 0). If ( x,y ) = (0 , 4), then the equation ( xy )( p + q ) = ( pq )( x + y ) reduces to 0( p + q ) = ( pq )4. So this means either p or q must also be zero and, then, it doesn’t matter what value we give to the other. 3. Graph isomorphism (a) Prove that the following two graphs are isomorphic. That is, for each vertex in G 1, give the corresponding vertex in G 2, making sure your mapping preserves the edge structure. G1: A B C D E G2: 1 5 3 4 2 Solution: A corresponds to 1, B corresponds to 5, C corresponds to 3, D corre-sponds to 4, and E corresponds to 2. (b) Prove that the following two graphs are not isomorphic. 2 G1: A B C D E G2: 1 2 3 4 5 Solution: They can’t be isomorphic because vertex 1 in G 2 has degree 4, but none of the vertices in G 1 has degree 4. 3...
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quiz3-review-solutions - x 0 If x,y =(0 4 then the equation...

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