Intersection Examples in Math

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

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 intersection of sets $A$ and $B$ is the set of all elements that belong to both $A$ and $B$ simultaneously, written $A \cap B$ .

Picture two overlapping circles in a Venn diagram—the intersection is only the overlapping region where both circles cover. For example, if set $A$ is students who play soccer and set $B$ is students who play piano, then $A \cap B$ is students who do both. It is the AND gate of set theory: an element must satisfy both conditions to be included.

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: The intersection keeps exactly the elements that belong to A and B simultaneously.

Common stuck point: The procedure for intersection is the easy part; the trap is including an element found in only one set. Asking "Does an item qualify only if it is in both sets at the same time?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does an item qualify only if it is in both sets at the same time?

Worked Examples

Example 1

easy

Let

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

and

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

. Find

A \cap B

Answer

A \cap B = \{3, 4\}

First step

Recall the definition:

A \cap B = \{x : x \in A \text{ and } x \in B\}

. An element belongs to the intersection only if it appears in both sets simultaneously.

Full solution

2
Check each element of $A = \{1,2,3,4\}$ : is $1 \in B$ ? No. Is $2 \in B$ ? No. Is $3 \in B = \{3,4,5,6\}$ ? Yes. Is $4 \in B$ ? Yes.
3
The elements common to both sets are 3 and 4, so $A \cap B = \{3,4\}$ .

The intersection keeps only the elements shared by both sets. It is always a subset of each original set.

Example 2

medium

Let

A = \{x \in \mathbb{R} : x \ge 1\}

and

B = \{x \in \mathbb{R} : x \le 4\}

. Find

A \cap B

Example 3

medium

Compute

\{x : 1 \le x \le 6\} \cap \{x : x \text{ is prime}\}

Example 4

hard

Prove

A \cap (A \cup B) = A

(absorption).

Practice Problems

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

Example 1

easy

Let

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

and

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

. Find

M \cap N

Example 2

easy

Find

A \cap B

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

and

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

Example 3

easy

Compute

A \cap B

for

A = \{1, 2, 3\}

and

B = \{2, 3, 4\}

Example 4

easy

Compute

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

Example 5

easy

Compute

A \cap A

for

A = \{5, 6\}

Example 6

easy

Compute

A \cap \emptyset

for

A = \{1, 2, 3\}

Example 7

easy

How many elements are in

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

Example 8

easy

Compute

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

Example 9

easy

A \cap B \subseteq A

always true?

Example 10

easy

Compute

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

Example 11

medium

Compute

(\{1, 2, 3\} \cap \{2, 3, 4\}) \cap \{3, 4, 5\}

Example 12

medium

A = \{x : x \text{ even}, 1\le x\le 10\}

and

B = \{x : x \text{ multiple of 3}, 1\le x\le 10\}

, find

A \cap B

Example 13

medium

Given

|A| = 7

|B| = 6

|A \cup B| = 10

, find

|A \cap B|

Example 14

medium

In a survey, 20 like coffee, 15 like tea, and 25 like at least one. How many like both?

Example 15

medium

A \cap B = A

, what relationship must hold between

A

and

B

Example 16

medium

Compute

A \cap B

where

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

and

B

is the set of odd numbers in

A

Example 17

medium

Two sets

A, B

are called disjoint when

A \cap B = \emptyset

. Are

\{1, 2\}

and

\{2, 3\}

disjoint?

Example 18

challenge

Prove that intersection is commutative:

A \cap B = B \cap A

Example 19

challenge

Prove the distributive law

A \cap (B \cup C) = (A \cap B) \cup (A \cap C)

Example 20

challenge

Sets

A, B, C

are pairwise disjoint with

|A|=3, |B|=4, |C|=5

. Find

|A \cup B \cup C|

and

|A \cap B \cap C|

Example 21

medium

Express 'students who like both math AND science' in set notation, given

M

and

S

Example 22

medium

List the intersection of 'multiples of 2' and 'multiples of 5' up to 20.

Example 23

easy

Compute

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

Example 24

easy

Compute

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

Example 25

easy

How many elements are in

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

Example 26

easy

Let

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

and

B=\{2,4\}

. Find

A \cap B

Example 27

medium

Let

E

= even integers,

P

= positive integers

\le 10

. Find

E \cap P

Example 28

medium

Compute

[1,5] \cap [3,8]

Example 29

medium

|A|=12

|B|=8

|A \cup B|=15

. Find

|A \cap B|

Example 30

medium

In a class of 30, 18 play soccer and 14 play basketball. If 25 play at least one, how many play both?

Example 31

medium

A \subseteq B

, what is

A \cap B

Example 32

medium

Compute

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

Example 33

medium

Compute

(-\infty, 3) \cap [0, \infty)

Example 34

medium

In a Venn diagram with two circles, the intersection is which region?

Example 35

hard

Let

A

= multiples of

4

from

1

100

;

B

= multiples of

6

from

1

100

. Find

|A \cap B|

Example 36

hard

Of 100 students, 40 like math, 50 like science, 30 like history. 20 like math and science; 15 like science and history; 10 like math and history; 5 like all three. How many like at least one subject?

Example 37

hard

Sets

A=\{x \in \mathbb{Z} : -5 \le x \le 5\}

and

B=\{x \in \mathbb{Z} : x^2 \le 9\}

. Find

A \cap B

Example 38

hard

De Morgan: rewrite

(A \cap B)^c

in terms of complements and union.

Example 39

hard

Compute

\bigcap_{n=1}^{\infty} \left[0, \tfrac{1}{n}\right]

Example 40

challenge

Prove that intersection distributes over union:

A \cap (B \cup C) = (A \cap B) \cup (A \cap C)

Example 41

challenge

Let

A_n = \{n, n+1, n+2, \ldots\}

for each

n \in \mathbb{N}

. Find

\bigcap_{n=1}^{\infty} A_n

Example 42

challenge

In a group of 200 people, 120 own a dog, 90 own a cat, 60 own a fish; 50 own dog and cat, 30 own dog and fish, 25 own cat and fish; 15 own all three. How many own none?

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Related Concepts

Set Union Venn Diagram

Background Knowledge

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

set