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Unformatted text preview: CS 181 — Winter 2007 Problem Set #0 Formal Languages and Automata Theory Optional Problem 0.1. (30 points) The power set P ( X ) of a set X is the set contain ing all subsets of X . For example, the power set of { a,b } is {∅ ,a,b, { a,b }} . It is well known that if the set X has n members then the power set of X has 2 n members. We’ll prove this inductively. Start by defining a new binary function ◦ called power set injection as follows: A ◦ c = { A ∪ { c }  A ⊆ A } Thus ◦ injects the possibly new element c into the powerset of A , so for example { a,b } ◦ c = {{ c } , { a,c } , { b,c } , { a,b,c }} 1. (5 points) What is the cardinality of A ◦ b when b 6∈ A ? You don’t need to prove your answer, but make sure it’s written in terms of the cardinality of the power set of A and not the cardinality of A . 2. (10 points) Write an inductive definition for power set using union and power set injection. The basis is simple: P ( X ) = {∅} iff X = ∅ What’s left is to define...
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This note was uploaded on 06/03/2008 for the course CS 181 taught by Professor Rupak during the Winter '08 term at UCLA.
 Winter '08
 Rupak

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