hw1-solutions - Solutions for Homework 1 of COM S 381 Fall...

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Solutions for Homework 1 of COM S 381 Fall 2006 September 4, 2006 Problem 1 (a) We want to give a one-one on-to mapping f from { 0 , 1 } * to the set of natural numbers. Define f ( ǫ ) = 1, f (0) = 2 , f (1) = 3 , f (00) = 4 , f (01) = 5 , f (10) = 6 , f (11) = 7 and so on. More precisely, for a string x ∈ { 0 , 1 } * , let | x | be the length of x and let number( x ) be the natural number x represents if we interpret it as a natural number written in binary. Define f ( x ) = 2 | x | + number( x ) . (1) This gives the desired one to one and on to mapping. (b) We will prove the claim by a contradiction. Let Σ = { 0 , 1 } . Since Σ * is countable, we can find a one-one onto function from N to Σ * (one example of such a function is the inverse of the function defined in the last part). If a string x is assigned to the natural number i , we call it the i -th string. We represent a subset S of strings by an infinite string s (called the characteristic string of the set S ), s [ i ] = 1 if the i -th string is in the set, s [ i ] = 0 otherwise. Please note that each infinite binary string represents a unique subset of Σ * and each subset has a string representing it. Therefore, the set of all subsets of Σ * is same as the set of all infinite binary strings. Let us assume that 2 Σ * is countable and f be a one-one onto function from N (the set of natural numbers) to 2 Σ * (represented by set of all infinite 1
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strings). We will show that there is an infinite string
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