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Finite Set: Definition & Overview

Finite Set: Definition & Overview
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  • 0:04 What is a Set?
  • 0:54 What is a Finite Set?
  • 1:23 Examples of Finite Sets
  • 2:52 Examples of Infinite Sets
  • 3:36 Sets as Elements of Other Sets
  • 4:42 Lesson Summary
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Lesson Transcript
Instructor: David Liano
After completing this lesson, you will be able to define what a finite set is and give examples of a finite set. You will also be able to explain the difference between a finite set and an infinite set.

What Is a Set?

Before we define a finite set, we need to establish what a set is. A set is a collection of elements or objects. Each element can be distinguished from other elements in the set. An example of a set would be all the prime numbers less than 20. Let's call this set A. Using proper set notation, we can write A as follows:

A = {2, 3, 5, 7, 11, 13, 17, 19}

Braces are typically used to enclose the elements of a set.

The cardinality of a set is the number of elements in the set. For instance, set A as defined earlier has a cardinality of 8 because it has eight elements. Using proper notation, we say that |A| = 8. The vertical bars indicate the cardinality of A.

What Is a Finite Set?

A finite set is a set that has a cardinality that equals a natural number (1, 2, 3…). Let's say that B represents a set. If |B| = n such that n is a natural number, then B is a finite set.

The empty set is also a finite set. The empty set has no elements and is denoted by the symbol Ø or by a pair of braces { }. The cardinality of the empty set is 0 (|Ø| = 0).

Examples of Finite Sets

The set described at the start of this lesson is an example of a finite set. Set A was defined as the prime numbers less than 20. There are eight prime numbers less than 20, so A had eight elements or a cardinality of 8.

Let's define a new set as follows:

B = (a, e, i, o, u)

B is the set of vowels in the English alphabet. B is a finite set because it has five elements or a cardinality of 5.

Let's define set C as the set of natural numbers less than 10. We can show C as follows:

C = {1, 2, 3, 4, 5, 6, 7, 8, 9}

We can also show C using an alternative form of set notation as shown in the following figure, which we will call Figure 1. The boldface capital N is often used to indicate the set of natural numbers. We can read the notation in Figure 1 as all elements x in the set of natural numbers such that x is less than 10.

Finite sets do not have to contain numbers and/or individual letters. For instance, the continents of the world can be a set as shown next:

{Africa, Antarctica, Asia, Australia, Europe, North America, South America}

This set of continents has a cardinality of 7.

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