16 Infinite Sets and Countability

16 Infinite Sets and Countability - Handout #16 April 16,...

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Handout #16 CS103 April 16, 2010 Robert Plummer Infinite Sets and Countability 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 that can be enumerated by the natural numbers from 1 up to some integer k. More precisely, set 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. A one-to-one correspondence between the elements of a set P and the elements of a set Q means that it is possible to pair off the elements of P and Q such that every element of P is paired off with a distinct element of Q. 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 or countably infinite . 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, etc.". More formally, a denumerable set is one where we can define a one-to-one correspondence between the elements in the set and the set of natural numbers, denoted N . Thus, the set N is, in a sense, the most basic of all infinite sets. 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 N . The counting scheme is the function that maps N to some other infinite set. Example 1 The set of positive even integers {2, 4, 6, 8, . ..} is denumerable because there is an obvious one-to-one correspondence between these integers and the natural numbers (f(n) = 2n). The set of all integers is denumerable because we can list them as follows: {0, 1, -1, 2, -2, 3, -3 . ..} and define the following one- to-one correspondence: f(n) = n/2 if n is even, -((n-1)/2) if n is odd. But at first blush, these examples seem to go against common sense. We might think to ourselves that the set of positive even integers must be 1/2 the size of the set of natural numbers and the set of all integers must be twice the size of the set of natural numbers. We have certainly defined functions for these two sets that are both one-to-one and onto, so by the basic definitions given above, N even and Z have the same cardinality as N . Surprising, but true!
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- 2 - Example 2 The set Q + (positive rational numbers) is denumerable. We know that each positive rational number can be written as a fraction p/q, where p and q are natural numbers. We can write all such fractions having the
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16 Infinite Sets and Countability - Handout #16 April 16,...

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