Venn diagrams and Set Notation

Lesson

Practice

Lesson

Venn Diagrams

A Venn Diagram is a pictorial way to display relationships between two different sets (or groups) of things. The idea of a Venn diagram was first introduced by John Venn in the late 1800's and they are still one of the most powerful visualisations for relationships.

Here is a simple example

Lets think about the numbers between $2$2 and $20$20.

I'm going to create two sets. Set $E=\left\{\text{even numbers}\right\}$E={even numbers}, and set $M=\left\{\text{multiplies of 3}\right\}$M={multiplies of 3}.

The next thing I do is write in all the numbers in the appropriate places. As I place a number I ask myself.... Is the number even? Is it a multiple of $3$3? Is it both? or Is it none of those options?

Take note of how the numbers that do not fit into either set are placed outside the circles, but still within the bounds of the universal set.

Now that we have a Venn diagram, we can answer a whole range of questions.

List the elements in events $E$E, $M$M and $E\cap M$E∩M

$E=\left\{2,4,6,8,10,12,14,16,18,20\right\}$E={2,4,6,8,10,12,14,16,18,20}

$M=\left\{3,6,9,12,15,18\right\}$M={3,6,9,12,15,18}

$E\cap M=\left\{6,12,18\right\}$E∩M={6,12,18}

Using Venn Diagrams when solving problems about sets will need us to be able to identify using set notation the regions in the Venn Diagram. The following applet will let you explore the different regions.

Created with Geogebra

Originally created by Mathguru

Worked Examples

Question 1

Consider the given Venn diagram.

Two overlapping circles create a Venn diagram with numerical values placed in various sections. The left circle is labeled "A" at the top, and the right circle is labeled "B" at the bottom. Inside the left circle, excluding the overlapping area, the numbers $2$ , $4$ , and $6$ are placed. Within the overlapping region, there are the numbers $7$ and $8$ . In the non-overlapping part of the right circle, the numbers $10$ and $14$ are placed. Below the left circle and outside of both circles, the number $16$ and $48$ is present.

State the elements that belong to $A\cap B$A∩B:
State the elements that belong to $A\cup B$A∪B:

QUESTION 2

Consider the given Venn diagram:

In the table below, match each numbered section of the Venn diagram with its description.

$A\cap B'\cap C'$`A`∩`B`′∩`C`′	$A\cap B\cap C'$`A`∩`B`∩`C`′	$A'\cap B\cap C'$`A`′∩`B`∩`C`′	$A\cap B'\cap C$`A`∩`B`′∩`C`	$A\cap B\cap C$`A`∩`B`∩`C`	$A'\cap B\cap C$`A`′∩`B`∩`C`	$A'\cap B'\cap C$`A`′∩`B`′∩`C`	$A'\cap B'\cap C'$`A`′∩`B`′∩`C`′
$\editable{}$	$\editable{}$	$\editable{}$	$\editable{}$	$\editable{}$	$\editable{}$	$\editable{}$	$\editable{}$

QUESTION 3

Consider the diagram below.

List all of the items in:

$A\cap C$A∩C
$\left(B\cap C\right)'$(B∩C)′
$A\cap B\cap C$A∩B∩C