Processing math: 100%
Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Mathematics LibreTexts

13.2: Properties

( \newcommand{\kernel}{\mathrm{null}\,}\)

The following facts outline some relationships countability and the set operations. They can be used to more easily prove that a set is countable or uncountable using the already-known countability or uncountability of a related set.

Proposition 13.2.1

  1. Every subset of N is countable.
  2. If there exists an injection AN, then the set A is countable.
  3. Suppose AB. If B is countable, then so is A.
  4. Suppose AB. If A is uncountable, then so is B.
  5. If A and B are countable, then AB and AB are both countable.

Example 13.2.3

The Cartesian product set R2=R×R is uncountable because it has an uncountable subset: the x-axis has the same size as R.


This page titled 13.2: Properties is shared under a GNU Free Documentation License 1.3 license and was authored, remixed, and/or curated by Jeremy Sylvestre via source content that was edited to the style and standards of the LibreTexts platform.

Support Center

How can we help?