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Section 1.2 Equivalent and Countable Sets

Section 1.2 Equivalent and Countable Sets - Section 1.2...

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Section 1.2: Equivalent and Countable Sets In essence, this section deals with the “size” of a given set. For example, it we consider the sets { } , , A a b c = and { } 1,2,3 B = , we would not say that the two sets are the same, but we could say that the sets are of the same “size.” On the other hand, let { } , , , , C α β δ ε λ = . In this case, we can say that A is “smaller” than C . We will make this notion of “size” more precise. Definition: Let A and B be sets. Then we say that A and B are equivalent , denoted A B , if there exists a one- to-one function f from A onto B . That is, if we can exhibit a function 1 1 : onto f A B - . Unsurprisingly, if A B it is also true that B A . Example: Let A and B be defined as above. Then define the function f from A to B by ( ) 1 f a = , ( ) 2 f b = , and ( ) 3 f c = . Then f is a bijection and so A B . On the other hand, while it is certainly possible to exhibit a function that maps one-to-one from A to C , it is impossible for this function to be onto. Hence, A C / . So far, this all seems very basic. The problem occurs when the sets we are considering are infinite. One might suspect that if A is a proper subset of B , then there is no way that A B ; and in fact, this is indeed the case if A is a finite set. However, as the next example will show, this is not the case with infinite sets. Example: The sets N and Z are equivalent. To see this, define the function : f N Z by 1 2 1 2 ( 1) if is odd ( ) if is even n n f n n n - = - We must show that f is indeed a bijection. Suppose ( ) ( ) f n f m = . Then it must be that n and m are both even or both odd. For if n is odd and m is even, then ( ) 0 f n and ( ) 0 f m < ; it follows that they could not possibly be equal in this case. Therefore, assume that n and m are both odd. It follows that 1 1 2 2 ( 1) ( 1) n m - = - . Some simple arithmetic shows that n m = ; a similar argument can be used in the case that both m and n are even. Hence, f maps one-to-one. Now, consider some integer z . Then either 0, 0, z z = or 0 z < . If 0 z = , let n = 1 and note that ( ) 0 f n z = = . If 0 z , let 2 1 n z = + . Again, ( ) f n z = . Finally, if 0 z < , let 2 n z = - , and once more, we find that ( ) f n z = . Thus, f maps onto Z and so f is a bijection from N to Z . We conclude that N Z . The question of the size of infinity and if there are different sized infinities is quite perplexing. To explore this question, we begin with the counting numbers, N . Sets that are equivalent to the set of positive integers deserve a special name. Notation: We will let m N denote the set { } 1,2,..., m m = N
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Definition: Let A be a set. Then A is finite if m A N ; A is countably infinite if A N . If A is either finite or countably infinite, we say that A is countable. Otherwise, we say that A is uncountable. Note that the example above shows that Z is countable. Alternatively, we can say that a set A is countable if and only if all of its elements can be arranged into a sequence (which in this case can be finite or infinite). What is meant here is that we can make a correspondence between N and A such as the one below N 1 2 3 4 5 6 7 8 L A 1 a 2 a 3 a 4 a 5 a 6 a 7 a 8 a L In this case, n a
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