MIT6_042JF10_assn06

MIT6_042JF10_assn06 - 6.042/18.062J Mathematics for...

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6.042/18.062J Mathematics for Computer Science October 13, 2010 Tom Leighton and Marten van Dijk Problem Set 6 Problem 1. [20 points] [15] For each of the following, either prove that it is an equivalence relation and state its equivalence classes, or give an example of why it is not an equivalence relation. (a) [5 pts] R n := { ( x,y ) Z × Z s.t. x y (mod n ) } (b) [5 pts] R := { ( x,y ) P × P s.t. x is taller than y } where P is the set of all people in the world today. (c) [5 pts] R := { ( x,y ) Z × Z s.t. gcd ( x,y ) = 1 } (d) [5 pts] R G := the set of ( x,y ) V × V such that V is the set of vertices of a graph G , and there is a path x,v 1 ,...,v k ,y from x to y along the edges of G . Problem 2. [20 points] Every function has some subset of these properties: injective surjective bijective Determine the properties of the functions below, and briefly explain your reasoning. (a) [5 pts] The function f : R R de±ned by f ( x ) = x sin( x ). (b) [5 pts] The function f : R R de±ned by f ( x ) = 99 x 99 . (c) [5 pts] The function f : R R de±ned by tan 1 ( x ). (d) [5 pts] The function f : N N de±ned by f ( x ) = the number of numbers that divide x . For example, f (6) = 4 because 1 , 2 , 3 , 6 all divide 6. Note: We defne here the set N to be the set oF all positive integers ( 1 , 2 ,... ) . Problem 3. [20 points] In this problem we study partial orders (posets). Recall that a weak partial order on a set X is reflexive ( x x ), anti-symmetric ( x y y
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This note was uploaded on 01/19/2012 for the course CS 6.042J / 1 taught by Professor Tomleighton,dr.martenvandijk during the Fall '10 term at MIT.

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MIT6_042JF10_assn06 - 6.042/18.062J Mathematics for...

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