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, and the right circle is labeled B. Inside circle A (left circle), excluding the overlapping area, are the numbers $2$2, $4$4, and $6$6. Inside circle B (right circle), excluding the overlapping area, are the numbers $10$10 and $14$14. Within the overlapping area are the numbers $7$7 and $8$8. Below and outside both circles are the numbers $16$16 and $48$48.

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:

In a rectangle, universal set $U$`U`, are three overlapping circles labeled $A$`A`, $B$`B`, and $C$`C` creating a Venn diagram. Circle $A$`A` is blue and contains the numbers $1$1, and $10$10, positioned outside the overlapping areas. Circle $B$`B` is pink, with the numbers $7$7, $12$12, and $14$14 placed outside the overlapping areas. Circle $C$`C` is black and contains the number $11$11 outside the overlapping areas. The section where only circles $A$`A` and $B$`B` overlap contains the number $3$3. The section where only circles $A$`A` and $C$`C` overlap contains the number $9$9. The section where only circles $B$`B` and $C$`C` overlap contains the numbers $6$6 and $8$8. The central overlap, where all three circles intersect, holds the numbers $4$4 and $5$5. The numbers $2$2 and $13$13 are located outside the circles but still within the boundary of the rectangle.

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