How to Define an Infinite Set

To fully understand how to define an infinite set, you need a good understanding of the meaning of each of the two terms individually. Once you have a good grasp of what "infinite" and "set" mean, defining an infinite set becomes easy and natural.

Instructions

Difficulty: Easy

1. Understand the meaning of the term "set." A set is any collection of objects or things. Think of a set of dishes, a set of tires or a set of encyclopedias. In mathematics, we often refer to sets of numbers. For example, the set of integers just means all the numbers that are integers. Another use of the term "set" in mathematics is the set of solutions for a particular problem. This refers to all the solutions that exist for the problem.
2. Understand the meaning of the term "infinite." There are several ways to describe the meaning of this term. Infinite means never-ending, boundless, unlimited, immeasurably large or going on and on without end. One good way to understand this meaning is to think of trying to count a number of objects. If you were to start counting now and count until the end of time, could you ever count all the objects? If not, then that group of objects is called infinite. The opposite of infinite is finite. If you can count all the objects in a group, that group is called finite.
3. Define an infinite set. Putting the meanings of both terms together as we understand them, an infinite set is simply a group of objects that could never be counted. The number of objects in an infinite set is not a definite number. Consider the set of tires we referred to in Step 1. Can you count how many tires are in any one set of tires? Sure--it's 4 for a standard automobile, 5 if we count the spare tire as part of the set. Could you ever count all the pieces in a set of dishes? It might take a while if the set is large, but, yes, we could count them. These are examples of finite sets--sets that have a definite number of objects.
4. Consider the set of counting numbers in mathematics. The counting numbers are, simply put, the numbers we use to count things. This set starts with the number 1, followed by 2, 3, 4 and so on. Could you ever count all these numbers? Nope, because these numbers go on forever; they never end. So we say the set of counting numbers is an infinite set. Another example of an infinite set is the set of all even numbers. If we started to name them--0, 2, 4, 6, 8, . . .--we could never name all of them. Infinite sets of numbers are often involved when working with mathematics, but not always. What about the set of all even numbers between 4 and 12? Can we name all of them? Yes, those numbers would be 6, 8 and 10. Since we can count, or name, all the members of the set, it is called a finite set.

Tips & Warnings

• True mathematicians beware! It should be obvious that the definitions and explanations given here are meant to facilitate understanding for the average person and are not mathematically precise.