08. Sets 2

08. Sets 2 - Maggie Johnson CS103B Handout #8 Infinite Sets...

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Maggie Johnson Handout #8 CS103B Infinite Sets and Countability Key topics: * Infinite Sets * Diagonalization * Russell's Paradox * The Beginnings of Theoretical Computer Science Infinite Sets In a finite set, we can always designate one element as the first member, another as the second member, etc. If there are k elements in the set, then these can be listed in the order we have selected: s 1 , s 2 , ..., s k Therefore, a finite set is one which can be enumerated by the positive integers from 1 up to some integer k. More precisely, A is finite if there is a positive integer k such that there is a one-to-one correspondence between A and the set of all natural numbers less than k. We define one-to-one correspondence between the elements of a set P and the elements of a set Q if it is possible to pair off the elements of of P and Q such that every element of P is paired off with a distinct element of Q. We are defining a function that maps the elements of P to the elements of Q; this function must be both one-to-one and onto (or bijective for you function enthusiasts out there). If a set is infinite, we may still be able to select a first element s 1 , and a second element s 2 , but we have no limit k. So the list of chosen elements may look like this: s 1 , s 2 , s 3 ...... Such an infinite set is called denumerable . Both finite and denumerable sets are countable sets because we can count, or enumerate the elements in the set. Being countable, however, does not always mean that we can give a value for the total number of elements in the set; it just means we can say "Here is a first one, here is a second one. ....". In other words, a denumerable set is one where we can define a one-to-one correspondence between the elements in the set and Z + . Thus, the set Z + is, in a sense, the most basic of all infinite sets. The reason for this is that the one- to-one correspondence can be used to "count" the elements of an infinite set. If f is a one-to-one and onto function from Z + to some infinite set A, then f(1) can be designated as the first element of A, f(2), the second, and so forth. Because f is one-to-one, no element is ever counted twice; and because f is onto, every element of A is counted eventually.
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To prove denumerability, we need only exhibit a counting scheme, i.e., if starting from a particular element, we can sequentially list all the elements in the list, (because such a listing will yield a one- to-one correspondence between the elements in the set and Z + ). The counting scheme is the function that maps Z + to some other infinite set. Example 1 The set of positive even integers {0, 2, 4, 6, 8. ..} is denumerable because there is an obvious one-to- one correspondence between these integers and the positive integers (2k). The set of all integers, positive and negative is denumerable because we can list them as follows: {0, 1, -1, 2, -2, 3, -3 . ..} (f(n) = n/2 if n is even, -(n-1/2) if n is odd). But these examples seem to go against common sense. The set of positive even integers must be 1/2 the size of the set of positive integers; the set of all
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08. Sets 2 - Maggie Johnson CS103B Handout #8 Infinite Sets...

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