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 }

A ∩ 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, 6 }

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