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 means it has a definite end you could reach; infinite means it goes on without bound.

Common stuck point: The procedure for finite vs infinite is the easy part; the trap is calling a huge set infinite. Asking "Could you, in principle, reach the last element — or does it never end?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Could you, in principle, reach the last element — or does it never end?

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.

Answer

(a) Finite; (b) Infinite; (c) Finite; (d) Finite.

First step

(a) Days of the week:

\{

Mon, Tue, Wed, Thu, Fri, Sat, Sun

\}

. Exactly

7

elements. Finite.

Full solution

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
(c) Letters in 'MATH': $\{M, A, T, H\}$ . Exactly $4$ elements. Finite.
4
(d) Decimal digits: $\{0, 1, 2, \ldots, 9\}$ . Exactly $10$ elements. 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.

Example 3

medium

A repeating decimal

0.\overline{142857}

has how many distinct decimal digits in its expansion, and is the digit string finite or infinite?

Example 4

medium

Finite or infinite: the set of solutions to

x + y = 10

over all real numbers?

Example 5

medium

Consider

S = \{n : n \text{ is a multiple of } 6 \text{ and } n \le 100\}

. Is

S

finite or infinite, and how many elements does it have?

Example 6

hard

Is the set of all polynomials with integer coefficients finite or infinite? Countable?

Example 7

hard

Hilbert's Hotel: an infinitely full hotel must take in a busload of

100

new guests. Describe a room reassignment that works.

Example 8

hard

Hilbert's Hotel: an infinitely full hotel must take in an infinite bus with one guest per natural number. Describe a room reassignment.

Example 9

hard

Classify and justify: (a) the number of grains of sand on every beach, (b) the number of decimal places needed to write

\pi

exactly.

Example 10

challenge

Show that the set of points

(x, y)

with

x, y

rational and

x^2 + y^2 = 1

is infinite.

Example 11

challenge

Is the set of subsets of

\mathbb{N}

(the power set) finite, countable, or uncountable?

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.

Example 3

easy

Is a jar holding

100

marbles a finite or infinite collection?

Example 4

easy

Are the counting numbers

1,2,3,\ldots

finite or infinite?

Example 5

easy

Is the set of grains of sand on Earth finite or infinite?

Example 6

easy

Finite or infinite: the number of points on a line segment from

0

1

Example 7

easy

Is the set of letters in the English alphabet finite or infinite?

Example 8

easy

Finite or infinite: the multiples of

5

, i.e.

5,10,15,\ldots

Example 9

easy

Is the number of even numbers between

1

and

10

finite or infinite?

Example 10

easy

Finite or infinite: the decimal digits of

\frac{1}{3}=0.333\ldots

Example 11

medium

Explain why 'a trillion elements' is still a finite set.

Example 12

medium

Is the set of primes finite or infinite? State the key fact.

Example 13

medium

Why might two infinite sets be the 'same size'? Give the integers and even integers as an example.

Example 14

medium

Classify and explain: the number of seconds in a day vs. the number of instants in a day.

Example 15

medium

Are all infinite sets the same size? Name two infinities of different sizes.

Example 16

medium

A sequence is defined by

a_n=\frac{1}{n}

for

n=1,2,3,\ldots

. Is the sequence finite or infinite, and is its set of values bounded?

Example 17

medium

Finite or infinite: solutions to

x^2=4

over the integers? Over the reals to

x^2\ge 0

Example 18

medium

Classify and explain: the set of fractions

\frac{1}{n}

for

n=1,2,3,\ldots

Example 19

medium

Is the set of whole numbers from

0

1{,}000{,}000

finite or infinite? How many elements?

Example 20

challenge

Prove that adding one element to an infinite countable set does not increase its size.

Example 21

challenge

Show the set of integers is the same size as the set of counting numbers by giving an explicit listing.

Example 22

challenge

Argue that the reals in

(0,1)

cannot be listed (sketch the diagonal idea).

Example 23

easy

Finite or infinite: the set of whole numbers less than

50

Example 24

easy

Finite or infinite: the set of integers

\mathbb{Z}

Example 25

easy

Finite or infinite: the set of perfect squares

1, 4, 9, 16, \ldots

Example 26

easy

How many elements are in the set

\{2, 4, 6, 8, 10\}

? Is it finite or infinite?

Example 27

medium

Finite or infinite: the set of rational numbers

\mathbb{Q}

? Is it countable?

Example 28

medium

Finite or infinite: the set of real numbers

\mathbb{R}

? Countable or uncountable?

Example 29

medium

Is the set of points inside a square of side

1

finite or infinite?

Example 30

medium

Finite or infinite: the set of solutions to

x^2 = 9

over the integers?

Example 31

medium

Finite or infinite: the set of solutions to

x + y = 10

over the positive integers?

Example 32

medium

How many primes are there? Finite or infinite? Name the theorem.

Example 33

hard

Is the union of two finite sets always finite? Justify.

Example 34

hard

The set of even naturals

\{2, 4, 6, \ldots\}

and the set of all naturals

\{1, 2, 3, \ldots\}

— which is bigger?

Example 35

hard

Is the set of all finite-length strings using letters

\{a, b\}

finite or infinite? Countable?

← Back to Finite vs Infinite Practice Mode

Related Concepts

Counting Set Cardinality Limit

Background Knowledge

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

countingset