MandysNotes

Sets of Numbers

20 March 2010 By In Basics & Number Properties
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A set is a collection of "elements," or "members," and each set is entirely determined by its members. If \[ x \]is a member of a set \[ U, \] we write \[ x \in U. \] In basic algebra, the elements of a set are usually numbers.

We can also designate a set by enclosing its members in  braces , {   }.

For example, if \[ U  \]is the set whose members are the numbers 1, 2, and 3, then: 

\[ U = \{1, 2, 3\}.  \]

Note that the order of the elements in the braces does not matter. Therefore:

\[ \{ 2, 1, 3 \} = \{3, 2, 1 \} = \{ 1, 2, 3 \} = U. \]

Any set with a finite number of members is a Finite set.

Any set whose members can be put in a one-to-one relationship with the set of natural numbers is called a Countably Infinite set.

A set that is either finite or countably infinite is called Countable.

An infinite set that cannot be put into a one-to-one relationship with the set of natural numbers is called an Uncountable set.

The set of natural numbers, the set of integers, and the set of rational numbers are examples of countable sets.

The set of irrational numbers and the set of real numbers are examples of uncountable sets.

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