Finite vs Infinite Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Finite vs Infinite.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Finite describes a quantity or set with a definite end; infinite describes something that goes on forever without bound.

A jar of 100 marbles is finite. The counting numbers are infinite.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Finite sets can be completely listed; infinite sets go on forever.

Common stuck point: Some infinite sets are 'bigger' than others (countable vs uncountable).

Sense of Study hint: Try listing out all the elements. If you can finish the list, it is finite. If the list requires '...' and never ends, it is infinite.

Worked Examples

Example 1

easy
Classify each set as finite or infinite and explain: (a) the days of the week, (b) the multiples of 7, (c) the letters in the word 'MATH', (d) the decimal digits.

Solution

  1. 1
    (a) Days of the week: \{Mon, Tue, Wed, Thu, Fri, Sat, Sun\}. Exactly 7 elements. Finite.
  2. 2
    (b) Multiples of 7: 7, 14, 21, 28, \ldots There is no largest multiple (given any multiple 7n, the next is 7(n+1)). Infinite.
  3. 3
    (c) Letters in 'MATH': \{M, A, T, H\}. Exactly 4 elements. Finite.
  4. 4
    (d) Decimal digits: \{0, 1, 2, \ldots, 9\}. Exactly 10 elements. Finite.

Answer

(a) Finite; (b) Infinite; (c) Finite; (d) Finite.
A set is finite if you can count all its elements and reach a final count. It is infinite if, no matter how many elements you list, there are always more. The multiples of any integer form an infinite set because multiplication produces new values without bound.

Example 2

medium
Is the set of decimal numbers between 0 and 1 finite or infinite? Is it countable or uncountable? Justify briefly.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
For each, say whether the process terminates (finite) or goes on forever (infinite): (a) counting to 1{,}000{,}000, (b) listing all prime numbers, (c) writing the decimal expansion of \frac{1}{3}.

Example 2

medium
A hotel has infinitely many rooms, all occupied. A new guest arrives. Explain (Hilbert's Hotel) how the hotel can accommodate the guest without anyone leaving.
← Back to Finite vs Infinite Practice Mode

Background Knowledge

These ideas may be useful before you work through the harder examples.

countingset