# midterm-2 - Fall 2009 CS131 Combinatorial Structures...

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Fall 2009 CS131 – Combinatorial Structures Midterm exam 2 Midterm exam 2 Only a single hand-written “crib” sheet can be used, no books or notes. Even if I ask for just a yes/no answer, you must always give a proof. You may get some points even if you write “I don’t know”, but if you write something that is wrong, you may get less. (It is not possible to pass just writing “I don’t know” everywhere. ...) Problem 1. Consider the set A = {1,2,3,4,5,6,7,8,9,10}, and the following relation R on A : we say that ( x , y ) R if 3 | x - y . (a) (5pts) Prove that this is an equivalence relation. Solution. ( x , y ) R if and only if x and y have the same remainder after division by 3. We have shown that such relations (deﬁned by the condition f ( x ) = f ( y ) for some function) are equivalence relations. (b) (5pts) List the elements of each equivalence class. Solution. The classes are {1,4,7,10} (remainder 1), {2,5,8} (remainder 2), {3,6,9} (remainder 0). Problem 2.

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## This note was uploaded on 10/15/2011 for the course MATHS 100 taught by Professor Fredphelps during the Fall '11 term at Jordan University of Science & Tech.

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midterm-2 - Fall 2009 CS131 Combinatorial Structures...

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