hw0 - CS 181 Winter 2007 Formal Languages and Automata...

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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 0 ∪ { c } | A 0 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 P ( X ) when X is non-empty. Here’s a hint: Recall that A = { a } can be written as A ∪ ∅ , B = { a, b
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