Can a countable set have an uncountable subset?

Any subset of a countable set is countable (eg. Z as a subset of Q). An uncountable set has both countable and uncountable subsets (eg. These sets are both uncountable (in fact, they have the same cardinality, which is also the cardinality of R, and R has infinite length).

How do you prove a set is countable or uncountable?

In mathematics, a set is countable if it has the same cardinality (the number of elements of the set) as some subset of the set of natural numbers N = {0, 1, 2, 3.}.

What does countably infinite mean?

Any set which can be put in a one-to-one correspondence with the natural numbers (or integers) so that a prescription can be given for identifying its members one at a time is called a countably infinite (or denumerably infinite) set.

What is the difference between a finite set uncountable set and countably infinite set?

Clearly every finite set is countable, but also some infinite sets are countable. Note that some places define countable as infinite and the above definition. In such cases we say that finite sets are “at most countable”. We say that a set A is uncountable if and only if it is not countable.

Are infinite sets countable or uncountable?

Not all infinite sets are the same. One way to distinguish between these sets is by asking if the set is countably infinite or not. In this way, we say that infinite sets are either countable or uncountable. We will consider several examples of infinite sets and determine which of these are uncountable.

How do you know if a set is uncountable?

If A is uncountable and B is any set, then the union A U B is also uncountable. If A is uncountable and B is any set, then the Cartesian product A x B is also uncountable. If A is infinite (even countably infinite) then the power set of A is uncountable.

What does it mean for a set to be countable?

This means that they can be put into a one-to-one correspondence with the natural numbers. The natural numbers, integers, and rational numbers are all countably infinite. Any union or intersection of countably infinite sets is also countable. The Cartesian product of any number of countable sets is countable.

Can the intersection of two infinite sets be countably infinite?

This shows that even countable intersection of countably infinite sets satisfying your property pairwise, can be countably infinite. Thanks for contributing an answer to Mathematics Stack Exchange!