## 2.1 Outline

- Denoting sets
- undefined term; set
- circular definition
- specifying sets
- description method
- roster method

- well-defined set
- terms
- members
- elements
- belong to
- contained in

- Sets of numbers
- natural numbers
- counting numbers
- integers
- rational numbers

- set-builder notation

- Equal and equivalent sets
- equal sets
- equivalent sets
- cardinality of a set
- cardinal number

- Universal and empty sets
- definitions
- notation for

- Venn diagrams
- complement
- general representation of a set

- Subsets and proper subsets
- definitions
- disjoint sets
- improper subset
- number of subsets
- general representation of two sets
- general representation of three sets

## 2.1 Essential Ideas

**Denoting Sets**

Sets are defined using the **description** or **roster methods**.

The objects in a set are called **members **or **elements** of the set.

The **cardinality** of a set is the number of elements in a set.

Two sets are **equal** if they contain the same number.

Two sets are **equivalent** if they have the same number of elements.

**Sets of Numbers**

**Natural Numbers**: {1, 2, 3, … }

**Whole Numbers**: {0, 1, 2, 3, … }

**Integers**: {…,, -2, -1, 0, 1, 2, … }

**Rational Numbers**: {*a/b *where *a* is an integer and *b* a nonzero integer}

**Special Sets**

The **universal set** contains all the elements under consideration in a given discussion.

The **empty set **contains no elements.

**Venn Diagrams**

One set divides the universe into 2 regions.

Two sets divide the universe into 4 regions.

Three sets divide the universe into 8 regions.

**Complement**

The complement of a set *S* is consists of everything that is not in *S*.

**Subsets**

A set *A* is a **subset** of a set *B*, if every element of *A* is also an element of *B*.

A set *A* is a **proper subset** of a set *B*, if every element of *A *is also an element of *B* **and** *A* and *B* are not equal sets.