# Intersection of sets

This lesson will explain how to find the intersection of sets. We will start with a definition of the intersection of two sets.

Definition:

Given two sets A and B, the intersection is the set that contains elements or objects that belong to A and to B at the same time.

We write A ∩ B

Basically, we find A ∩ B by looking for all the elements A and B have in common.  Next, we illustrate with examples.

Example #1.

To make it easy, notice that what they have in common is in bold.

Let A = { 1 orange, 1 pineapple, 1 banana, 1 apple } and B = { 1 spoon, 1 orange, 1 knife, 1 fork, 1 apple }

A ∩ B = { 1 orange, 1 apple }

Example #2.

Find the intersection of A and B and then make a Venn diagrams.

A = { b, 1, 2, 4, 6 } and B = { 4, a, b, c, d, f }

∩ B = { 4, b } Example #3.

A = { x / x is a number bigger than 4 and smaller than 8 }

B = { x / x is a positive number smaller than 7 }

A = { 5, 6, 7 } and B = { 1, 2, 3, 4, 5, }

A ∩ B = { 5, 6 }

Or A ∩ B = { x / x is a number bigger than 4 and smaller than 7 }

Example #4.

A = { x / x is a country in Asia }

B = { x / x is a country in Africa }

Since no countries in Asia and Africa are the same, the intersection is empty.

A ∩ B = { }

Example #5. A = { #, %, &, *, \$ }

B = { }

This example is subtle! Since the empty set is included in any set, it is also included in A although you don't see it.

Therefore, the empty set is the only thing set A and set B have in common.

A ∩ B = { }

In fact, since the empty set is included in any set, the intersection of the empty set with any set is the empty set.

Definition of the union of three sets:

Given three sets A, B, and C the intersection is the set that contains elements or objects that belong to A, B, and to C at the same time.

We write A ∩ B ∩ C

Basically, we find A ∩ B ∩ C by looking for all the elements A, B, and C have in common.

A = { #, 1, 2, 4, 6 }, B = { #, a, b, 4, c } and C = A = { #, %, &, *, \$, 4 }

A ∩ B ∩ C = { 4 , # }

The graph below shows the shaded region for the intersection of two sets The graph below shows the shaded region for the intersection of three sets This ends the lesson about intersection of sets. If you have any questions about the intersection of sets, I will be more than happy to answer them.

## Recent Articles 1. ### Introduction to Physics

Nov 18, 20 01:20 PM

Top-notch introduction to physics. One stop resource to a deep understanding of important concepts in physics

New math lessons

Your email is safe with us. We will only use it to inform you about new math lessons.

Tough Algebra Word Problems.

If you can solve these problems with no help, you must be a genius! Real Life Math Skills

Learn about investing money, budgeting your money, paying taxes, mortgage loans, and even the math involved in playing baseball.