Practice Cardinality in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick 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.

Showing a random 20 of 50 problems.

Example 1

medium

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

Example 2

easy

Find

|\{\emptyset\}|

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

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 5

medium

From

\{1,\ldots,20\}

, how many are divisible by

4

Example 6

hard

How many subsets of

\{1,\ldots,10\}

contain exactly

4

elements with no two consecutive?

Example 7

hard

From

\{1,\ldots,200\}

, how many integers are divisible by

3

5

Example 8

easy

A set

S

has 3 elements. How many subsets does

S

have? How many proper subsets?

Example 9

challenge

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

Example 10

easy

A=\{a,b,c\}

, how many subsets does

A

have?

Example 11

medium

How many bijections exist from a

5

-element set to itself?

Example 12

easy

Find

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

using the convention

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

Example 13

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 14

medium

How many 3-element subsets does

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

have?

Example 15

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\}\}

Example 16

easy

Find

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

Example 17

easy

A=\{a,b,c\}

and

B=\{d,e\}

are disjoint, find

|A\cup B|

Example 18

easy

|\{a,b,c\}|

a number or a set?

Example 19

challenge

Cantor showed

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

. What is the name for the cardinality of

\mathbb{R}

Example 20

hard

In a survey of

100

60

read books,

50

watch films,

30

do both. How many do exactly one?

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Related Concepts

Set Element