1. What Is a Set?
A set is a well-defined collection of distinct objects.
The objects contained in a set are called elements or members.
Sets are usually represented by capital letters such as: A, B, C, S
For example: $$A=\{1,2,3,4\}$$
This means that the set A contains the elements 1, 2, 3, and 4.
2. Membership of a Set
The symbol
$$\in$$
means “is an element of.”
For example: $$3\in A$$
means that 3 belongs to set A.
The symbol
$$\notin$$
means “is not an element of.”
For example: $$7\notin A$$
3. Writing a Set
A set can be represented in different ways.
Roster Form
List all elements inside braces.
$$A=\{2,4,6,8\}$$
Set-Builder Form
Describe the condition that elements must satisfy.
$$A=\{x\mid x\text{ is an even positive integer less than }10\}$$
Both expressions describe the same set.
4. Order and Repetition
The order of elements does not matter.
Therefore: $$\{1,2,3\}=\{3,2,1\}$$
Repeating an element also does not create a new element.
Therefore: $$\{1,1,2,2,3\}=\{1,2,3\}$$
5. Important Number Sets
Students should recognize the following notation.
Natural numbers
$$\mathbb{N}$$
Integers
$$\mathbb{Z}$$
Rational numbers
$$\mathbb{Q}$$
Real numbers
$$\mathbb{R}$$
A useful relationship is:
$$\mathbb{N}\subseteq\mathbb{Z}\subseteq\mathbb{Q}\subseteq\mathbb{R}$$
We will study the subset symbol in more detail in the next lesson.
6. Empty Set
A set containing no elements is called the empty set.
It can be written as: $$\emptyset$$
or $$\{\}$$
Example:
The set of real numbers satisfying
$$x^2=-1$$
is empty : $$\emptyset$$
7. Universal Set
The universal set contains all objects being considered in a particular problem.
It is commonly represented by:
$$U$$
For example, if: $$U=\{1,2,3,4,5,6\}$$
and $$A=\{2,4,6\}$$
then every element of AA belongs to the universal set U.