MIT6_042JF10_rec11_sol

# MIT6_042JF10_rec11_sol - 6.042/18.062J Mathematics for...

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6.042/18.062J Mathematics for Computer Science October 15, 2010 Tom Leighton and Marten van Dijk Notes for Recitation 11 1. Give a description of the equivalence classes associated with each of the following equivalence relations. (a) Integers x and y are equivalent if x y (mod 3). Solution. { ..., 6 , 3 , 0 , 3 , 6 ,... } { ..., 5 , 2 , 1 , 4 , 7 ,... } { ..., 4 , 1 , 2 , 5 , 8 ,... } (b) Real numbers x and y are equivalent if x ± = y ± , where z ± denotes the smallest integer greater than or equal to z . Solution. For each integer n , all the real numbers r such that n 1 < r n form an equivalence class. 2. Show that neither of the following relations is an equivalence relation by identifying a missing property (reﬂexivity, symmetry, or transitivity). (a) The “divides” relation on the positive integers. Solution. This relation is reﬂexive (since a | a ) and transitive (since a | b and b | c implies a | c ), but not symmetric (since 3 | 6, but not 6 | 3). (b)

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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_rec11_sol - 6.042/18.062J Mathematics for...

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