Set Notation & Venn Diagrams

 

What Is Set Notation?

A set is simply a group of elements. These elements could be numbers, letters, or even outcomes of a probability experiment.

A set is written with curly brackets: $\{2, 4, 6, 8\} \quad \text{(the set of even numbers from 1 to 10)}$

If there's a rule for the set, use a colon: $\{x : x < 5\} \quad \text{means all values of } x \text{ less than 5}$

This can also be written with a vertical bar: $\{x \mid x < 5\}$

Special Notation

• $\mathcal{E}$: The universal set (everything we're considering)
• $A \cap B$: Intersection of A and B (shared elements)
• $A \cup B$: Union of A and B (all elements from A, B, or both)
• $A'$: The complement of A (everything in E\mathcal{E} not in A)
• $x \in A$: x is an element of A

 

Venn Diagrams

A Venn diagram visually shows sets and how they interact inside a rectangle representing the universal set $\mathcal{E}$.

Each circle represents a set. Where circles overlap, you have intersections.

• $A \cap B$: Region where A and B overlap

• $A \cup B$: All regions within A or B

 

 

• $A'$: Everything outside of A

 

 

Example

In a group of 40 students:

• 25 study History (H)
• 18 study Geography (G)
• 10 study both History and Geography
• 5 study neither

(a) Draw a Venn diagram to represent this information.

Start with the overlap: 10 students are in both H and G.

• History only: 25 - 10 = 15
• Geography only: 18 - 10 = 8
• Neither: 5

Now total: 15 (H only) + 10 (both) + 8 (G only) + 5 (neither) = 38 students ✓

So the Venn diagram looks like:

 

 

Example

Question: Use the Venn diagram above to find:

(i) The probability that a student studies only History.

There are 15 students who study History but not Geography.

Total students = 40

$P(\text{History only}) = \frac{15}{40} = \frac{3}{8}$

(ii) The probability that a student studies at least one subject.

All students except those who study neither:

$P(\text{History or Geography}) = 1 - \frac{5}{40} = \frac{35}{40} = \frac{7}{8}$

(iii) The probability that a student studies History given that they study Geography.

This is a conditional probability:

Let’s count how many students study Geography: 10 (both) + 8 (G only) = 18

Number of those who also study History = 10

$P(H \mid G) = \frac{10}{18} = \frac{5}{9}$