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CS 341 Automata Theory Elaine Rich Homework 1 Due Thursday, September 7 at 11:00 a. m. 1) Write each of the following explicitly: a) P ({ a , b }) – P ({ a , c }) b) { a , b } × {1, 2, 3} × c) { x : ( x 7 x 7} d) { x : 5 y ( y < 10 ( y + 2 = x ))} (where is the set of nonnegative integers) e) { x : 5 y ( 5 z   (( x = y + z ) ( y < 5) ( z < 4)))} 2) Give an example, other than one of the ones in the book, of a relation on the set of people that is reflexive and symmetric but not transitive. 3) Define p to be “equivalent modulo p ”. x p y iff x modulo p = y mod p . For example 4 3 7 and 7 5 12. Let R p be a binary relation on , defined as follows, for any p 1: R p = {( a , b ): a p b } So, for example R 3 contains (0, 0), (6, 9), (1, 4), etc., but does not contain (0, 1), (3, 4), etc. a) Is R p an equivalence relation for every p 1? Prove your answer. b) If R p is an equivalence relation, how many equivalence classes does R p induce for a given value of p ? What are they? (Any concise description is fine.) 4) Are the following sets closed under the following operations? If not, give an example that proves that they are not and then specify what the closure is. a) The negative integers under division b) The positive integers under exponentiation c) The finite sets under Cartesian product 5) Give examples to show that the intersection of two countably infinite sets can be either finite or countably infinite, and that the intersection of two uncountable sets can be finite, countably infinite, or uncountable.
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