Finite Sets and Infinite Sets

Read this chapter to learn the concepts of finite and infinite sets, their definitions, and properties. Finite sets are countable and contain a finite number of elements, while infinite sets are not finite and extend to infinity. Examples of countable infinite sets include rational numbers.

We encounter both finite and infinite sets in our daily lives, and this chapter aims to provide a better understanding of their differences using examples. It will also provide examples to illustrate the differences between finite and infinite sets.

What is Finite Sets?

Finite sets, also known as countable sets, have a finite number of elements that can be counted. They end with a starting and ending element. These sets can be represented in roster notation form, such as the set of vowels in the English alphabets V = {a, e, I, o, u}, which has a finite number of elements.

All finite sets are countable, but not all countable sets are finite. For instance, set for a set with odd natural numbers less than 19, and A = {3, 5, 7, 9, 11} it is a valid set with 5 elements which is countable.

Some of properties of finite sets are listed below:

• We can get a proper subset of a finite set which is also finite.
• The union of any finite sets is also finite.
• The intersection of two finite sets is also finite.
• The cartesian product of finite sets is finite.
• The cardinality of a finite signifies finite number and it is equal to the number of elements in that set.
• Similarly, the power set of a finite set is also finite.

What is Infinite Sets?

Infinite sets are not finite, meaning their elements are infinite. They are infinite from start to end or both sides have continuity. Examples include the set of whole numbers and the set of real numbers. These sets are uncountable and their elements are represented by dots: W = {0, 1, 2, 3, ..}, representing the infinity of the set. The set of whole numbers is an example of an infinite set.

An infinite set has an element count that is infinite, value that we are unable to pinpoint the precise number of elements. On the other hand, countable infinite sets have countable element counts. One such example is the set of integers Z = { -2, -1, 0, 1, 2, .}, which has an infinite element count and whose elements can be put in a one-to-one correspondence with the set of natural numbers.

Some of the important properties of infinite sets include the following −

• The union of any infinite sets is also infinite set.
• The power set of infinite set is also infinite.
• The superset of an infinite set returns infinite set.
• Sometimes a subset of an infinite set is finite or sometimes infinite.
• Infinite sets can be both countable and uncountable.

Venn Diagram of Finite Sets and Infinite Sets

As we know, a Venn diagram is used to represent relationships among sets or groups of objects. The following figure shows the relationship between finite and infinite sets.

A finite set consists of elements 1, 15, and 20, while an infinite set consists of natural numbers and whole numbers. Multiple finite sets can be created from an infinite set, as seen in the example of a finite set lying inside an infinite set.

Difference between Finite Sets and Infinite Sets

The following table compares and contrasts the features of Finite and Infinite Sets −

Conclusion

In this chapter, we explained the concepts of finite sets and infinite sets. Finite sets have elements that we can count, whereas infinite sets have elements we cannot count because they have infinitely many elements.

We described the properties with examples along with the differences and Venn diagram for a better understanding.