7 - Infinite Sets
Summary
Equinumerous Sets
In this chapter, we'll discuss a method of comparing the sizes of infinite sets. Surprisingly, we'll find that, in a sense, infinity comes in different sizes! By now, you should be fairly proficient at reading and writing proofs, so we'll give less discussion of the strategy behind proofs and leave more proofs as exercises.
For finite sets, we determine the size of a set by counting. What does it mean to count the number of elements in a set? When you count the elements in a set A, you point to the elements of A in turn while saying the words one, two, and so forth. We could think of this process as defining a function f from the set {1, 2, …, n} to A, for some natural number n. For each i ∈ {1, 2, …, n}, we let f(i) be the element of A you're pointing to when you say “i.” Because every element of A gets pointed to exactly once, the function f is one-to-one and onto. Thus, counting the elements of A is simply a method of establishing a one-to-one correspondence between the sets {1, 2, …, n} and A, for some natural number n.
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- How to Prove ItA Structured Approach, pp. 306 - 328Publisher: Cambridge University PressPrint publication year: 2006