Subset Examples in Math

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

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

Concept Recap

Set $A$ is a subset of set $B$ if every element of $A$ is also an element of $B$ , written $A \subseteq B$ .

Every single thing in $A$ can also be found inside $B$ . Think of $A$ as fitting entirely within $B$ , like a small circle inside a big one.

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 is a subset of B when every single member of A is also a member of B.

Common stuck point: The procedure for subset is the easy part; the trap is declaring $A \subseteq B$ after finding just one common element. Asking "Is every single member of the first set also a member of the second?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is every single member of the first set also a member of the second?

Worked Examples

Example 1

easy

Let

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

and

B = \{2, 4\}

. Determine whether

B \subseteq A

Answer

B \subseteq A

First step

Recall the definition:

B \subseteq A

means every element of

B

is also an element of

A

. We check each element of

B

individually.

Full solution

2
Check $2 \in B$ : is $2 \in A = \{1,2,3,4,5\}$ ? Yes. Check $4 \in B$ : is $4 \in A$ ? Yes.
3
Since every element of $B$ belongs to $A$ , we conclude $B \subseteq A$ . Note also $B \ne A$ since $A$ has elements (1, 3, 5) not in $B$ , so $B$ is a proper subset: $B \subsetneq A$ .

B

is a subset of

A

if every element of

B

belongs to

A

. We verify this by checking membership one element at a time.

Example 2

medium

List all subsets of

S = \{a, b, c\}

Example 3

medium

List all subsets of

\{1, 2\}

that contain

1

Example 4

medium

List all proper subsets of

\{a, b\}

Example 5

medium

Given

A = \{x : x \text{ is even, } 0 < x < 10\}

and

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

, is

A = B

Example 6

hard

How many subsets of

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

have size exactly

3

Practice Problems

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

Example 1

easy

Let

X = \{1, 2, 3\}

and

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

. Is

X \subseteq Y

? Is

Y \subseteq X

Example 2

easy

Let

A = \{1, 3\}

and

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

. Is

A \subseteq B

? Is

B \subseteq A

Example 3

easy

A = \{1, 2\}

a subset of

B = \{1, 2, 3\}

Example 4

easy

Is the empty set

\emptyset

a subset of

\{1, 2, 3\}

Example 5

easy

A = \{1, 2, 3\}

a subset of itself?

Example 6

easy

\{4\}

a subset of

\{1, 2, 3\}

Example 7

easy

Distinguish: is the statement

\{1\} \subseteq \{1, 2\}

1 \subseteq \{1, 2\}

correct?

Example 8

easy

\{2, 3\}

a subset of

\{1, 2, 3, 4\}

Example 9

easy

\{1, 2, 3\}

a subset of

\{1, 2\}

Example 10

easy

True or false: every element of a set

A

is also a subset of

A

Example 11

medium

How many subsets does the set

\{a, b\}

have? List them.

Example 12

medium

How many subsets does

\{1, 2, 3\}

have?

Example 13

medium

A = \{1, 2\}

a PROPER subset of

B = \{1, 2\}

Example 14

medium

List all subsets of

\{1, 2, 3\}

that contain the element

1

Example 15

medium

A \subseteq B

and

B \subseteq C

, with

A=\{1\}

B=\{1,2\}

C=\{1,2,3\}

, is

A \subseteq C

Example 16

medium

\{1, 2, 3\}

a subset of

\{1, 2, 3, 3\}

Example 17

medium

How many subsets does the empty set

\emptyset

have?

Example 18

challenge

Prove that the empty set

\emptyset

is a subset of every set

A

Example 19

challenge

A set

A

has

|A| = n

. Prove it has exactly

2^n

subsets.

Example 20

challenge

A \subseteq B

, prove that

|A| \le |B|

for finite sets.

Example 21

medium

Given

A = \{1, 2\}

, is

\{1, 2\} \in

the power set of

A

, and is

\{1, 2\} \subseteq A

Example 22

medium

\{1, 2\}

a proper subset of

\{1, 2, 3\}

Example 23

easy

\{5\} \subseteq \{1, 5, 9\}

Example 24

easy

\emptyset \subseteq \emptyset

Example 25

easy

How many subsets does

\{x\}

have?

Example 26

easy

Distinguish: is

1 \in \{1, 2, 3\}

1 \subseteq \{1, 2, 3\}

Example 27

easy

Is the set of even integers a subset of the set of integers?

Example 28

medium

How many subsets of

\{1, 2, 3, 4\}

are there?

Example 29

medium

\{2, 4, 6\}

a proper subset of

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

Example 30

medium

A \subseteq B

and

|B| = 5

, what is the maximum possible

|A|

Example 31

medium

True or false:

\{1, 2\} \subseteq \{\{1, 2\}, 3\}

Example 32

medium

How many subsets of

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

contain

3

Example 33

medium

\{1, 1, 2\}

a subset of

\{1, 2, 3\}

? (note: sets ignore duplicates)

Example 34

medium

\mathbb{N} \subseteq \mathbb{Z}

? Is

\mathbb{Z} \subseteq \mathbb{N}

Example 35

medium

A \subseteq B

and

B \subseteq A

, what can you conclude?

Example 36

hard

Let

A = \{1, 2\}

. How many subsets does the power set

\mathcal{P}(A)

have?

Example 37

hard

Show: if

A \subseteq B

and

B \subseteq C

, then

A \subseteq C

(transitivity).

Example 38

hard

How many subsets of

\{1, 2, \dots, 10\}

contain at least one odd number?

Example 39

hard

Is the set of primes a subset of the set of natural numbers?

Example 40

challenge

Let

A = \{1, 2, 3\}

. How many ordered pairs

(X, Y)

of subsets of

A

satisfy

X \subseteq Y

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Related Concepts

Set Element

Background Knowledge

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

setelement