# Set theory

## 1 About

## 2 Inequalities (numbers)

If \(a\) and \(b\) are numbers, then exactly one of the following is true:

- \(a < b\)
- \(a > b\)
- \(a = b\)

This is called the **law of trichotomy**.

## 3 Inequalities (sets)

Sets do not obey the law of trichotomy. For example, none of the following are true:

- \(\{1\} \subset \{2\}\)
- \(\{1\} \supset \{2\}\)
- \(\{1\} = \{2\}\)

## 4 Here’s a feature

Spaces seem not to matter when you are in math mode. But they do seem to matter when in delimiting of math mode. [Most users will not understand this previous sentence.]

- \(\{1\}\subset\{2\}\)
- $ {1} {2}$

## 5 Union

[This contains the minimum needed to teach the Latex.]

\[ X = A \cup B \]

- Words: X equals A union B
- Latex:
`X = A \cup B`

## 6 Intersection

[This contains the minimum needed to teach the math.]

The intersection of two sets \(A\) and \(B\) are the objects \(x\) that are members of \(A\) and a member of \(B\). The intersection is written as:

\[ X = A \cap B \]

- Example: \(\{1, 3, 5, 7\} \cap \{3, 5, 9\} = \{3, 5\}\).
- Example: \(3 \in A\) and \(3 \in B\) so \(3 \in (A\cap B)\).