8.1-PreClass.pdf - Chapter 8 Cardinality In this chapter,...

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Chapter 8CardinalityIn this chapter, we will explore the notion of cardinality, which formalizes what it meansfor two sets to be the same “size”.8.1Introduction to CardinalityWhat does it mean for two sets to have the same “size”? If the sets are finite, this is easy:just count how many elements are in each set. Another approach would be to pair up theelements in each set and see if there are any left over. In other words, check to see if thereis a one-to-one correspondence (i.e., bijection) between the two sets.But what if the sets are infinite? For example, consider the set of natural numbersNand the set of even natural numbers 2N:={2n|n2N}. Clearly, 2Nis a proper subset ofN. Moreover, both sets are infinite. In this case, you might be thinking thatNis “largerthan” 2NHowever, it turns out that there is a one-to-one correspondence between thesetwo sets. In particular, consider the functionf:N!2Ndefined viaf(n) = 2n. It is easilyverified thatfis both one-to-one and onto. In this case, mathematics has determinedthat the right viewpoint is thatNand 2Ndo have the same “size”. However, it is clear

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Term
Spring
Professor
N/A
Tags
Natural number, Finite set, Bijection

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