hw1-sol

# hw1-sol - ECS 120 Introduction to the Theory of Computation...

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ECS 120: Introduction to the Theory of Computation Homework 1 Due Apr 8, by 1pm in the homework box in Kemper 2131 Problem 1. Let A, B, C be three sets. Prove the following: (a) A \ B = A \ ( B A ). (b) B A if and only if A B = . (c) ( A \ B ) \ C =( A \ C ) \ ( B \ C )= A \ ( B C ). (d) A B = and A B = if and only if A = B . (a) Let x be any element from the set A \ B .Then x A and x 6∈ B .Thu s x 6∈ A B , since it is not in both. But then x A \ ( B A ). So, all elements of A \ B are elements of A \ ( B A )too . Conv e r se ly ,le t x A \ ( B A ). Then, x A and x 6∈ A B s x is in A but not B .S o , x A \ B and therefore all elements of A \ ( B A ) are elements of A \ B too. (b) (If) If A B = and x B ,then x A , i.e. B A . (Only if) If B A and x B then x A . Then, x 6∈ A , and thus A B = . (c) First we show that A \ B = A B .Name ,i f x A and x 6∈ B then x A and x B .T h u s x A B .C o n v e r s e l y f x A B then x A and x B h u s x A and x 6∈ B , and the result follows.

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hw1-sol - ECS 120 Introduction to the Theory of Computation...

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