probability_extra.pdf

# probability_extra.pdf - Countable and Uncountable Sets In...

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Countable and Uncountable Sets In this section we extend the idea of the “size” of a set to infinite sets. It may come as somewhat of a surprise that there are di ff erent “sizes” of infinite sets. At the end of this section we show that there are infinitely many di ff erent such “sizes”. For the most part we focus on a classification of sets into two categories: the sets whose elements can be listed (countable sets) and those for which there is no list containing all of the elements of the set (uncountable sets). Definition (sets can be put into 1-1 correspondence) . Let A and B be sets and f : A B be a 1-1 correspondence. Then, by Propositions F12 and F13 in the Functions section, f is invertible and f 1 is a 1-1 correspondence from B to A . Because of the symmety of this situation, we say that A and B can be put into 1-1 correspondence . It is easy to see that any two finite sets with the same number of elements can be put into 1-1 correspondence. Conversely, if A and B are finite sets that can be put into 1-1 correspondence, then we know from Corollary F6 in the section on Functions that A and B have the same number of elements. Definition (two sets have the same cardinality, cardinality n ) . We say that two sets A and B have the same cardinality (or size), and write | A | = | B | , if they can be put into 1-1 correspondence. We say that a set A has cardinality n , and write | A | = n , if there exists a natural number n such that A can be put into 1-1 correspondence with { 1 , 2 , . . . , n } . The cardinality of a finite set is just the number of elements it contains (by Corollary F6), so the definition agrees with previous use of the notation | A | . (Notice that { 1 , 2 , . . . , n } is empty when n = 0.) Notice that the above definition opens the door to talking about infinite sets having the same cardinality (when they can be put into 1-1 correspondence), or not (when they can’t). Thus, the definition allows for the possiblity that not all infinite sets have the same cardinality. This turns out to be true, but it will take some time before we have developed enough results and techniques to demonstrate it. Example (infinite sets having the same cardinality) . Let S = { n 2 : n Z + } . The function f : Z + S defined by f ( n ) = n 2 is a 1-1 correspondence. ( Exercise : prove it.) Therefore, | S | = | Z + | . Thus the set S of squares of positive integers is the same “size” as the set Z + of positive integers. This may seem confusing since S Z + . Such a situation can not arise for finite sets, it is a consequence of the sets being infinite. One definition of an infinite set is a set which can be put into 1-1 correspondence with a proper subset of itself. For aother example the formula f ( n ) = n + 1 is a 1-1 correspondence from N to Z + . (This implies | N | = | Z + | .) Exercise . Prove that | Z | = | Z + | = | E | , where E = { 0 , 2 , 4 , 6 , . . . } is the set of even positive integers. Hint: one way to do this is for the function f : Z + Z to be defined according to two cases – the even positive integers map to the non-negative integers, and the odd positive integers map to the negative integers.

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• Fall '17
• BenReichardt
• Natural number, Countable set

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