Overview:
The mathematical concept of infinity involves sets of numbers that grow past any limit, no matter how huge that limit is. One-to-one correspondence is a useful tool for comparing sets of numbers and points. There seem to be levels and sizes to infinite sets, so that some infinite sets are larger than others.
What Are Infinite Sets?
Many different sets of numbers exist that are infinite, as they have no end. The set of integers is infinite. On other words, for every number that there is, there is another even larger in the set. Similarly, the set of even numbers {2, 4, 6, 8 …} is infinite, as is the set of odd numbers {1, 3, 5, 7…}.
How Does One-to-One Correspondence Work with Infinite Sets?
One-to-one correspondence means that every number in one set can be paired with a unique number in another set. For example, the set of even numbers {2, 4, 6, 8…} corresponds to the set of odd numbers {1, 3, 5, 7…}. The number 1 is paired with 2, 3 is paired with 4, 5 is paired with 6, 7 is paired with 8, and so on. Even though the individual numbers in each set are not countable, they can be paired in one-to-one correspondence.
What about Cardinality?
One of the paradoxes of infinite sets is that many infinite sets can be paired by one-to-one correspondence, even though one set contains the members of another set. In other words, they have the same cardinality. For example, the set of natural numbers {1, 2, 3, 4…} and the set of even numbers {2, 4, 6, 8…} can be paired in one to one correspondence. The number 1 from the first set is paired with 2 from the second set, the number 2 from the first set is paired with 4 from the second set, the number 3 from the first set is paired with 6 from the second set, so that n from the first set is paired with 2n in the second set. Suppose another set {1, 2, 3, 4…} is paired with {2, 3, 4, 5…}. The two sets still have the same cardinality, as 1 can be paired with 2, 2 with 3, 3 with 4, and so on, so that n from the first set is paired with n + 1 from the second set.
Are There Different Sizes of Infinite Sets?
Not all infinite sets have the same cardinality. A mathematician named Georg Cantor showed in the late 19th century that the set of real numbers (rational numbers and irrational numbers) is larger than the set of integers, using one-to-one correspondence. Other mathematicians since Cantor have postulated even larger sets of numbers and mathematical entities.
Interested in algebra tutoring services? Learn more about how we are assisting thousands of students each academic year.
<span class=”tutorOrange”>SchoolTutoring Academy</span>is the premier educational services company for K-12 and college students. We offer tutoring programs for students in K-12, AP classes, and college. To learn more about how we help parents and students in St. John’s, NL, Canada: visit Tutoring in St. John’s, NL, Canada