Set Examples in Math

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

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

Concept Recap

A well-defined collection of distinct, unordered objects called elements, described either by listing or by a membership rule.

Think of a set as a bag that can hold anything — numbers, names, shapes — but with two strict rules: no duplicates allowed and the order in which items sit inside the bag does not matter.

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: A set is a collection where duplicates collapse to one and rearranging changes nothing.

Common stuck point: The procedure for set is the easy part; the trap is counting a repeated entry twice, like calling $\{a, a, b\}$ a 3-element set. Asking "If I rearrange the items or drop a repeat, is it still the exact same object?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: If I rearrange the items or drop a repeat, is it still the exact same object?

Worked Examples

Example 1

easy

Let

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

. Determine whether

6 \in A

and whether

5 \in A

Answer

6 \in A \text{ and } 5 \notin A

First step

Recall that a set is a well-defined collection of distinct objects called elements. The set

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

is given in roster (list) notation.

Full solution

2
Check membership of 6: scan the roster — 2, 4, **6**, 8, 10. The element 6 appears, so $6 \in A$ .
3
Check membership of 5: scan the roster — 2, 4, 6, 8, 10. The element 5 does not appear, so $5 \notin A$ .

The symbol

\in

means 'is an element of.' To test membership, check whether the element appears in the set's roster.

Example 2

medium

Write the set

B = \{x \in \mathbb{Z} : -2 \le x < 3\}

in roster notation.

Example 3

medium

Convert

\{x \in \mathbb{Z} : 1 \le x \le 6 \text{ and } x \text{ is odd}\}

to roster form.

Example 4

medium

Write the set

D

of letters in the word "banana".

Example 5

hard

Write the set of solutions in

\mathbb{R}

x^2 - 5x + 6 = 0

Practice Problems

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

Example 1

easy

Let

C = \{a, e, i, o, u\}

. List all elements of

C

and state the cardinality

|C|

Example 2

easy

Are the sets

\{2, 4, 4, 6\}

and

\{6, 2, 4\}

equal? Explain.

Example 3

easy

\{1, 2, 3\}

the same set as

\{3, 2, 1\}

Example 4

easy

How many elements are in the set

\{1, 1, 2, 3\}

Example 5

easy

\{3\}

the same thing as the number

3

Example 6

easy

Write the set of even numbers between 1 and 9 by listing.

Example 7

easy

5

an element of the set

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

Example 8

easy

Does the set

\{a, b, c\}

change if we rewrite it as

\{b, c, a\}

Example 9

easy

How many elements does the empty set

\emptyset

have?

Example 10

easy

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

Example 11

medium

Set

A

is described as 'all integers

x

with

x^2 = 9

'. List

A

by roster.

Example 12

medium

How many elements are in

\{x : x \text{ is a letter in the word } \textbf{LEVEL}\}

Example 13

medium

\{\emptyset\}

the same as

\emptyset

Example 14

medium

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

, list the elements of

A

that satisfy

x > 3

Example 15

medium

A set has elements

\{a, \{b\}, c\}

. How many elements does it have, and is

b

one of them?

Example 16

medium

Are the sets

\{x : x \text{ is even}\}

and

\{2k : k \in \mathbb{Z}\}

equal?

Example 17

medium

List the set of prime numbers less than 10.

Example 18

challenge

Prove that if sets

A

and

B

satisfy

A \subseteq B

and

B \subseteq A

, then

A = B

Example 19

challenge

Set

S

contains its own number of elements rule:

S = \{1, 2, \dots, n\}

has exactly

n

elements. If a set

T

satisfies

|T| = |T \cup \{5\}|

, what must be true about

5

Example 20

challenge

Explain why

\{1, 2\}

and

\{1, 2, 1\}

have the same number of elements, then give the cardinality.

Example 21

medium

Is the collection 'all tall people' a well-defined set?

Example 22

medium

Is the collection of 'the three best movies ever' a well-defined set?

Example 23

easy

Let

A = \{0, 1, 2, 3\}

. Is

0 \in A

? Is

-1 \in A

Example 24

medium

Are the sets

A = \{1, 2, 3\}

and

B = \{x \in \mathbb{Z} : 0 < x < 4\}

equal?

Example 25

easy

Is the empty set the same as

\{0\}

Example 26

medium

List the subset of

\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}

consisting of perfect squares.

Example 27

medium

True or false:

\{1, 2, 3\}

has the same elements as

\{2, 3, 1, 2\}

Example 28

hard

Write the set of positive integer divisors of

12

Example 29

hard

Let

A = \{1, 2\}

. List all subsets of

A

Example 30

medium

Let

E = \{x \in \mathbb{Z} : 0 \le x \le 8 \text{ and } x \text{ is even}\}

. Find

E

and

|E|

Example 31

medium

\{x \in \mathbb{N} : x < 0\}

the empty set?

Example 32

hard

Let

A = \{2, 3, 5, 7\}

and

B = \{1, 4, 6\}

. List the elements of

A

that are NOT in

B

Example 33

medium

Write the set of integers

n

with

|n| \le 2

in roster form.

Example 34

hard

Two coaches list players: Coach A lists

\{

Maya, Ben, Liu, Tara

\}

; Coach B lists

\{

Ben, Liu, Tara, Maya

\}

. Are the two team sets equal?

Example 35

challenge

How many distinct

2

-element subsets does

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

have?

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Related Concepts

Element Subset Union Intersection