Definition: Sets are not given a formal definition. The are collections of things, or lists of things, in which nothing is repeated.

There are two notations for sets.

• One way is to list all of the elements of the set, or list elements in a pattern followed by \(\dots\) For example, the set of numbers \(1\) through \(4\), can be written \(\{1, 2, 3, 4\}\). The set of numbers 1 and higher can be written \(\{1, 2, 3, \dots\}\).
• A second way is to write a set as a rule. For example, to write the set of all of the real numbers from 1 to 4, use \(\{x : 1 \leq x \leq 4\}\). This set includes numbers like \(\pi\) since \(\pi\) is between 1 and 4. We will modify this later.

Order does not matter, so \(\{1, 2, 3, 4\} = \{4, 2, 1, 3\}\)

Elements cannot be repeated, so \(\{1, 1, 2\}\) is not a set.

Sometimes sets are defined by description. For example, let \(S\) be the set of all people who live in Iowa, or let \(B\) be the set of all black holes. Although I can't list all of the elements in the sets or find a pattern to describe the sets, I can tell you whether something is an element of the set or not.

You can think of the set \(A\) as the red circle.

The set \(A\) is the set of points \(\{1, 5, 7\}\).