Cardinality Examples in Math

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

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

Concept Recap

The cardinality of a finite set is the number of distinct elements it contains, written $|A|$ — it measures the size of the set without regard to element order or identity.

Cardinality answers "how many?" — count each distinct element once and you have the cardinality.

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: Cardinality is the count of distinct elements in a set, written $|A|$ .

Common stuck point: The procedure for cardinality is the easy part; the trap is counting a repeated listing twice, like $|\{a, a, b\}| = 3$ . Asking "Am I counting how many distinct elements a set has, each once?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I counting how many distinct elements a set has, each once?

Worked Examples

Example 1

easy

Find the cardinality of: (a)

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

, (b)

B = \{x \in \mathbb{N} : x \le 0\}

, (c)

C = \{\{1,2\}, 3, \{4\}\}

Answer

|A|=5,\quad |B|=1,\quad |C|=3

First step

(a) Count the distinct elements:

2, 4, 6, 8, 10

— five elements, so

|A| = 5

Full solution

2
(b) The only natural number $\le 0$ is $0$ (assuming $0 \in \mathbb{N}$ ). So $B = \{0\}$ and $|B| = 1$ .
3
(c) $C$ has three elements: the set $\{1,2\}$ , the number $3$ , and the set $\{4\}$ . So $|C| = 3$ .

Cardinality counts distinct top-level elements. When a set contains other sets as elements, each sub-set counts as one element regardless of its own size.

Example 2

medium

Let

A = \{1,2,3,4\}

and

B = \{3,4,5,6\}

. Verify the formula

|A \cup B| = |A| + |B| - |A \cap B|

Example 3

medium

At a party of

60

35

like jazz,

40

like rock,

20

like both. How many like at least one genre?

Example 4

hard

Show that

|\mathbb{Z}|=|\mathbb{N}|

by giving a bijection.

Practice Problems

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

Example 1

easy

A set

S

has 3 elements. How many subsets does

S

have? How many proper subsets?

Example 2

medium

In a class of 40 students, 25 play football, 20 play basketball, and 10 play both. Use cardinality formulas to find how many play at least one sport and how many play neither.

Example 3

easy

Find

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

Example 4

easy

Find

|\{1,1,2\}|

Example 5

easy

Find

|\emptyset|

Example 6

easy

Find

|\{\emptyset\}|

Example 7

easy

A=\{a,b,c\}

and

B=\{d,e\}

are disjoint, find

|A\cup B|

Example 8

easy

How many subsets does a 3-element set have?

Example 9

easy

Find

|\{x \in \mathbb{Z} : 1 \le x \le 5\}|

Example 10

easy

|\{a,b,c\}|

a number or a set?

Example 11

medium

|A|=7

|B|=5

|A\cap B|=3

, find

|A\cup B|

by inclusion-exclusion.

Example 12

medium

In a class, 18 study French, 15 study Spanish, 6 study both. How many study at least one?

Example 13

medium

Find

|A\cup B|

|A|=10

|B|=8

, and

A,B

are disjoint, then if

|A\cap B|=4

Example 14

medium

How many subsets of

\{1,2,3,4\}

have exactly 2 elements?

Example 15

medium

|A|=m

and

|B|=n

, what is

|A\times B|

(the Cartesian product)?

Example 16

medium

Among 30 students, 20 like tea, 16 like coffee, all like at least one. How many like both?

Example 17

medium

How many subsets does

\{1,2,3,4,5\}

have, and how many are proper subsets?

Example 18

medium

How many functions are there from a 3-element set to a 2-element set?

Example 19

medium

Of 40 people, 22 own a dog, 18 own a cat, 10 own neither. How many own both?

Example 20

challenge

Use inclusion-exclusion for three sets:

|A|=|B|=|C|=10

, each pairwise intersection

=4

, triple intersection

=2

. Find

|A\cup B\cup C|

Example 21

challenge

How many elements are in

\{1,2,\dots,100\}

that are divisible by 2 or 3?

Example 22

challenge

Show

|\mathbb{N}| = |\{\text{even naturals}\}|

by exhibiting a bijection.

Example 23

easy

Find

|\{a,b,c,d,e,f\}|

Example 24

easy

Find

|\{x\in\mathbb{Z}:-2\le x\le 2\}|

Example 25

easy

A=\{a,b,c\}

, how many subsets does

A

have?

Example 26

easy

Find

|\{\{1,2\},\{3\},4\}|

Example 27

easy

|\mathbb{N}|

finite or infinite?

Example 28

easy

Find

|\{x\in\mathbb{N}:x<6\}|

using the convention

\mathbb{N}=\{1,2,3,\ldots\}

Example 29

medium

|A|=12

|B|=8

, and

|A\cup B|=15

, find

|A\cap B|

Example 30

medium

A class of

50

students:

30

take math,

25

take physics,

10

take both. How many take neither?

Example 31

medium

How many 3-element subsets does

\{1,2,3,4,5,6\}

have?

Example 32

medium

|A|=4

and

|B|=5

, find

|A\times B|

Example 33

medium

How many functions are there from a

4

-element set to a

3

-element set?

Example 34

medium

How many bijections exist from a

5

-element set to itself?

Example 35

medium

From

\{1,\ldots,20\}

, how many are divisible by

4

Example 36

hard

In a survey of

100

60

read books,

50

watch films,

30

do both. How many do exactly one?

Example 37

hard

From

\{1,\ldots,200\}

, how many integers are divisible by

3

5

Example 38

hard

Three sets

|A|=|B|=|C|=20

, pairwise intersections

=8

, triple intersection

=3

. Find

|A\cup B\cup C|

Example 39

hard

How many

4

-letter strings use only letters from

\{A,B,C\}

with repeats allowed?

Example 40

hard

How many subsets of

\{1,\ldots,10\}

contain exactly

4

elements with no two consecutive?

Example 41

challenge

|\mathbb{Q}|=|\mathbb{N}|

Example 42

challenge

Cantor showed

|\mathbb{N}|<|\mathbb{R}|

. What is the name for the cardinality of

\mathbb{R}

Example 43

challenge

How many subsets of

\{1,\ldots,20\}

have even cardinality?

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Related Concepts

Set Element

Background Knowledge

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

setelement