Countably infinite
Definition
A set is countably infinite if its elements can be put into a one-to-one correspondence with the natural numbers. This means there is a way to list the elements of the set in a sequence without missing any.
5 Must Know Facts For Your Next Test
- The set of all natural numbers (N) is an example of a countably infinite set.
- A set that is countably infinite has the same cardinality as the set of natural numbers.
- The set of rational numbers (Q) is also countably infinite.
- Functions or sequences can be used to show that two sets are in one-to-one correspondence.
- Any subset of a countably infinite set that is also infinite will itself be countably infinite.
Review Questions
- What does it mean for a set to be countably infinite?
- Give an example of a countably infinite set other than the natural numbers.
- How can you demonstrate that two sets are in one-to-one correspondence?
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Related terms
Cardinality: The measure of the number of elements in a set.
Natural Numbers: The set of positive integers starting from 1, often denoted by N.
One-to-One Correspondence: A relationship between two sets where each element of one set pairs with exactly one element of another, and vice versa.
