How can you tell if an infinite set is countable or uncountable?

How can you tell if an infinite set is countable or uncountable?

A set is called countable, if it is finite or countably infinite. Thus the sets are countable, but the sets are uncountable. Any subset of a countable set is countable. Any infinite subset of a countably infinite set is countably infinite.

How do you show set is finite?

Corollary 9.8. A finite set is not equivalent to any of its proper subsets. Let B be a finite set and assume that A is a proper subset of B. Since A is a proper subset of B, there exists an element x in B−A.

What makes a set uncountable?

A set is uncountable if it contains so many elements that they cannot be put in one-to-one correspondence with the set of natural numbers. Uncountable is in contrast to countably infinite or countable. For example, the set of real numbers in the interval [0,1] is uncountable.

Which of the following are countable sets?

The sets N, Z, the set of all odd natural numbers, and the set of all even natural numbers are examples of sets that are countable and countably infinite.

When an infinite set is finite?

An infinite set is endless from the start or end, but both the side could have continuity unlike in Finite set where both start and end elements are there. If a set has the unlimited number of elements, then it is infinite and if the elements are countable then it is finite.

Which of the following is not finite set?

A set which is not finite is called an infinite set. Now we will discuss about the examples of finite sets and infinite sets. Then, P is a finite set and n(P) = 6.

Is a subset of an uncountable set uncountable?

If a set has a subset that is uncountable, then the entire set must be uncountable. These sets are both uncountable (in fact, they have the same cardinality, which is also the cardinality of R, and R has infinite length). So by rearranging an uncountable set of numbers you can obtain a set of any length what so ever!

How do you show uncountable sets?

The most common way that uncountable sets are introduced is in considering the interval (0, 1) of real numbers. From this fact, and the one-to-one function f( x ) = bx + a. it is a straightforward corollary to show that any interval (a, b) of real numbers is uncountably infinite.

What is the difference between countable and uncountable sets?

Sets such as N and Z are called countable, but “bigger” sets such as R are called uncountable. The difference between the two types is that you can list the elements of a countable set A, i.e., you can write A = { a 1, a 2, ⋯ }, but you cannot list the elements in an uncountable set. For example, you can write N = { 1, 2, 3, ⋯ },

When is a Union of finite sets countable?

Here, we are making use of the very simple principle that a countable union of finite sets is countable. To put this more formally, if is a set that can be written as a union for some collection of finite sets, then is countable.

What is the intuition behind the countable subset theorem?

The intuition behind this theorem is the following: If a set is countable, then any “smaller” set should also be countable, so a subset of a countable set should be countable as well. To provide a proof, we can argue in the following way.

How do you prove that a set is countable?

Here is a quick informal account of a standard proof that the set is countable: we can list its elements in the order , , , , , , , and so on. (This is informal because I have talked about “lists” and have not actually defined how the sequence continues.)