7.03 Sets and Venn diagrams

A set is a collection of objects that have a common property. So, if you think about all the things in your pencil case, this could be considered a set.

You could list many other sets of objects, such as he clothes you have in your wardrobe, the names of streets you pass by on your way to school, animals that have four legs or people in your family.

To describe a set using mathematical notation, we use large curly brackets and list all the items of the set between them. Each object in the set is called an element. The set of objects shown in the picture above can be written down as:

$\left\{\text{pencils, pens, sharpener, protractor, scissors, eraser, compass, glue, highlighter, calculator}\right\}${pencils, pens, sharpener, protractor, scissors, eraser, compass, glue, highlighter, calculator}

Of course we can have sets in mathematics as well, and these sets tend to have numbers or algebraic symbols.

The set of odd numbers less than $10$10 would look like this: $\left\{1,3,5,7,9\right\}${1,3,5,7,9}

The set of multiples of $5$5 up to $50$50: $\left\{5,10,15,20,25,30,35,40,45,50\right\}${5,10,15,20,25,30,35,40,45,50}

The set of positive factors of $24$24: $\left\{1,2,3,4,6,8,12,24\right\}${1,2,3,4,6,8,12,24}

It is helpful to know that mathematicians call a set with no elements in it an empty set.

Intersections and unions

Just like how a road intersection is the place where two roads cross paths, an intersection of sets is where two sets overlap. Elements that appear in the intersection of sets are elements that have the same characteristic as both the individual sets.

Mathematically we write the intersection of sets using the intersection symbol, $\cap$∩. We interpret the intersection of $A$A and $B$B, $A\cap B$A∩B to be what appears in both set $A$A and set $B$B. It helps some students to relate $\cap$∩ to 'AND' or to think of the symbol like a bridge joining both sets.

For example, given the two numerical sets $A=\left\{5,10,15,20,25,30\right\}$A={5,10,15,20,25,30} and $B=\left\{6,12,18,24,30\right\}$B={6,12,18,24,30} then we can find the intersection $A\cap B=\left\{30\right\}$A∩B={30}.

If we consider the intersection the AND of mathematical sets, then the union is the 'OR'. $A\cup B$A∪B is the notation we use, and we would read this as either the union of $A$A and $B$B. The solution is the list of all of the elements that lie in $A$A or $B$B.

For example, given two sets $A=\left\{5,11,16,17,20,25\right\}$A={5,11,16,17,20,25} and $B=\left\{4,12,15,25,30\right\}$B={4,12,15,25,30} we find the union to be $A\cup B=\left\{4,5,11,12,15,16,17,20,25,30\right\}$A∪B={4,5,11,12,15,16,17,20,25,30}.

Visualising multiple sets using Venn diagrams

We are often required to determine whether the information given to us in a practical probability problem involves sets with overlap. As well as using the language and notation of sets as described above, it can be helpful to do this visually.

Imagine you have the following clothes in your wardrobe. This is our universal set, which simply means that it is the set of everything relevant to this question.

From this we could create two sets, the set $P=\left\{\text{shirts in your wardrobe}\right\}$P={shirts in your wardrobe}

and the set $Q=\left\{\text{blue clothes in your wardrobe}\right\}$Q={blue clothes in your wardrobe}

We now can display this in a Venn Diagram, which is made utilising overlapping circles. The idea of a Venn diagram was first introduced by John Venn in the late 1800s and they are still one of the most powerful visualisations for relationships. Imagine you are sorting the objects into two groups.

So we begin sorting our clothes, which is fairly straightforward.

The blue shirt corresponds to both sets, so it is put into the centre overlapping section of the diagram.

Worked example

Example 1

Let's think about the numbers between $2$2 and $20$20.

We are 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 to do is write in all the numbers in the appropriate places. As we place a number we consider - is the number even? Is it a multiple of $3$3? Is it both, or is it neither 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 range of questions.

We can 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}

We can also find the complement of a set, which is every element in the universal set not inside a given set:

$\overline{M}$M$=\left\{1,2,4,5,7,8,10,11,13,14,16,17,19\right\}$={1,2,4,5,7,8,10,11,13,14,16,17,19}

$\overline{E\cup M}$E∪M$=\left\{5,7,11,13,17,19\right\}$={5,7,11,13,17,19}

As you can see, different set notation corresponds to different regions in a Venn diagram. The following applet will let you explore the different regions. (Note that in this applet, the complement is notated using $A'$A′ instead of $\overline{A}$A.)

Created with Geogebra Originally created by Mathguru

 

Practice questions

Question 1

QUESTION 2

QUESTION 3

QUESTION 4