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Infinity is infinitely large, right? But is it possible for one infinite set to be larger than another?
Infinity is an important property of the real number system and its subsets. Let's consider the relative magnitude of the infinite sets of numbers we just diagrammed and explore whether infinity can come in more than one size. Note 3
One definition of infinity says that a set is infinite if it can be put into one-to-one correspondence with a proper subset of itself. (A proper subset is one that is missing at least one element of the original set.) In one-to-one correspondence, each element in the first set matches exactly one element in the second set, and vice versa (i.e., each element in the second set matches exactly one element in the first set).
For example, if we have two finite sets, {1, 2, 3} and {a, b, c}, they can be put into one-to-one correspondence in the following way: 1 is paired with a, 2 is paired with b, and 3 is paired with c (and vice versa, a is paired with 1, b is paired with 2, and c is paired with 3). Here's another way to demonstrate this correspondence:
Now let's set up a similar numeric relationship between the counting numbers and the even counting numbers to show that the two sets can be put into one-to-one correspondence with each other:
By placing the numbers like this, we can see that for each element n in the set of counting numbers, there is a one-to-one corresponding element 2n in the even numbers set, and vice versa. This pattern extends infinitely.
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