Infinite Set — Definition, Formula & Examples
An infinite set is a set that has no last element — its members go on without end, so you can never finish listing all of them. The set of natural numbers {1, 2, 3, …} is the most common example.
A set is infinite if there exists no non-negative integer such that can be put into a one-to-one correspondence with the set . Equivalently, is infinite if and only if it can be placed in a one-to-one correspondence with a proper subset of itself.
How It Works
To determine whether a set is infinite, ask: can I pair every element with exactly one natural number and eventually run out of elements? If you never run out, the set is infinite. Some infinite sets are countably infinite, meaning their elements can be listed in a sequence (like the integers). Others are uncountably infinite, meaning no such listing is possible (like the real numbers between 0 and 1). This distinction, discovered by Georg Cantor, shows that not all infinities are the same size.
Example
Problem: Determine whether the set of even positive integers E = {2, 4, 6, 8, …} is infinite.
Step 1: Try to match each element of E with a natural number n using the rule f(n) = 2n.
Step 2: For every natural number n, there is a corresponding even number 2n, and this mapping never terminates. No finite count exhausts all elements of E.
Step 3: Notice that E is a proper subset of the natural numbers, yet we just showed a one-to-one correspondence between them. By the formal definition, this confirms E is infinite.
Answer: E = {2, 4, 6, 8, …} is an infinite set (specifically, countably infinite).
Why It Matters
Infinite sets are foundational to calculus, where you work with the real number line — an uncountably infinite set. Understanding the distinction between finite and infinite sets also prepares you for topics in discrete mathematics and computer science, such as computability theory, where the question of whether certain problems can be solved depends on the nature of infinite sets.
Common Mistakes
Mistake: Assuming all infinite sets are the same size.
Correction: Cantor's theorem proves that the set of real numbers is strictly larger than the set of natural numbers. Infinite sets can have different cardinalities.