basic mathematics image
 basic mathematics image

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 Ç B

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


Example #1.


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

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

A Ç 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 Ç B = {4, b}

Venn-diagrams-image


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 Ç B = {5, 6}

Or A Ç 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 Ç B = { }

Example #5.


Sets-image


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 Ç 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 Ç B Ç C

Basically, we find A Ç B Ç 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 Ç B Ç C = {4 , # }

The graph below shows the shaded region for the intersection of two sets

Intersection-image


The graph below shows the shaded region for the intersection of three sets

Intersection-of-three-sets-image


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.








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