Essential Prerequisites: Sets and Set Notation in College Algebra

Unit 1: Essential Prerequisites

Introduction to Sets and Set Notation

Understanding sets and their notation is a foundational skill in college algebra and higher mathematics. This section introduces the concept of a set, how sets are defined, and the notation used to describe them. Mastery of these ideas is essential for later topics such as functions and relations.

Definition of a Set

• Set: A set is a well-defined collection of elements. The term "well-defined" means that there is no ambiguity about whether an object belongs to the set.

• Example of a Not Well-Defined Set: The set of all "small apples in a barrel" is not well-defined, because the criteria for "small" are unclear.

• Example of a Well-Defined Set: The set of all positive integers is well-defined, as there is no ambiguity about which numbers are included.

Set Notation

• Sets are typically denoted using curly braces { }.

• Example: The set S of digits from 0 to 9 is written as: $S = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$

• Each element of the set is listed within the braces, separated by commas.

Types of Sets

• Finite Set: A set with a limited (countable) number of elements. Example: $S = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$ is a finite set.

• Infinite Set: A set with an unlimited (uncountable) number of elements. Example: The set of all positive integers can be written as: $P = \{1, 2, 3, 4, 5, 6, \ldots\}$

• Empty Set: A set with no elements. It is denoted by the symbol $\emptyset$ or $\{\}$. Example: If $E$ is the empty set, then $E = \emptyset$.

Summary Table: Types of Sets

Type of Set	Definition	Example
Finite Set	Contains a limited number of elements	$\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$
Infinite Set	Contains an unlimited number of elements	$\{1, 2, 3, 4, 5, 6, \ldots\}$
Empty Set	Contains no elements	$\emptyset$ or $\{\}$

Key Points

• Well-defined sets are crucial for clarity in mathematics.

• Set notation uses curly braces and commas to list elements.

• Sets can be finite, infinite, or empty.

Example Applications

• Finite Set Example: The set of vowels in the English alphabet: $V = \{a, e, i, o, u\}$

• Infinite Set Example: The set of all even numbers: $E = \{2, 4, 6, 8, \ldots\}$

• Empty Set Example: The set of real numbers less than 0 and greater than 1: $S = \{x \mid x < 0 \text{ and } x > 1\} = \emptyset$