# Venn Diagrams

**Venn diagrams**, also called **set** diagrams, are helpful in calculating **probabilities**.

Symbols used include:

- ∪ = union of sets
- ∩ = intersection of sets
- A = complement of set A (the elements that are not in set A)
- ∅ = null or empty set

## Example One

The Venn diagram shows two sets called set A and set B:

**Elements of set A** = { 1, 2, 3, 4, 10 }
**Elements of set B** = { 4, 5, 6, 7, 9 }

**A ∪ B** is the **union of sets A and B**.

A ∪ B = { 1, 2, 3, 4, 10, 5, 6, 7, 9 }

**A ∩ B** is the **intersection (overlap) of sets A and B**.

A ∩ B = { 4 }

**A** is the **complement of set A**, that is, the elements that are **not in set A**.

A = { 5, 6, 7, 9, 8 }

**B** is the **complement of B**, that is, the elements that are **not in set B**.

B = { 1, 2, 3, 4, 10, 8 }

**(A ∪ B)** is the **complement of A ∪ B**, that is, the elements that are **not in A ∪ B**.

(A ∪ B) = { 8 }

**(A ∩ B)** is the **complement of A ∩ B**, that is, the elements that are **not in A ∩ B**.

(A ∩ B) = { 1, 2, 3, 10, 5, 6, 7, 9, 8 }

*Note that the elements can be listed in any order.*

## Example Two - Social Networking

The Venn diagram shows the names of students who use social networking sites of **Facebook (set F) and Twitter (set T)**:

Elements of set F = { Al, Bob, Courtney, Freda }

Elements of set T = { Bob, Dave, Ernie, Gary }

**Q1.** Which students are in F ∪ T (the union of sets F and T)?
**Q2.** Which students are in A ∩ B (the intersection of sets F and T)?
**Q3.** Who is in (F ∪ T) and does not use Facebook or Twitter?

**Q4.** What is the probability that a student uses only Facebook?
**Q5.** What is the probability that a student does not use any social network?

**Answers:**

**A1.** F ∪ T = { Al, Bob, Courtney, Freda, Dave, Ernie, Gary }

**A2.** F ∩ T = { Bob }

**A3.** (F ∪ T) = { Harry }

**A4.** Probability of using Facebook only = ^{3}⁄_{8}

**A5.** Probability of not using social network sites = ^{1}⁄_{8}

## Example Three - Cyber-Bullying

The Venn diagram shows the names of 12 students in a class at school. Following incidents of cyber-bullying, the teacher drew this Venn diagram of students who are involved in cyber-bullying as **Cyber-bullies (set B)** and/or **Cyber-victims (set V)** and the **bullying or victimization has to the teacher (set T)**.

**Q1.** Which students are bullies?
**Q2.** Which students are victims?
**Q3.** Who is both bully and victim?
**Q4.** Which bullies have been reported to the teacher?
**Q5.** Which victims have been reported to the teacher?
**Q6.** Which students who are both bullies and victims have not been reported to the teacher?
**Q7.** What is Kerry's role in these incidents?
**Q8.** Which students have no involvement at all?
**Q9.** What is the probability that a student in this class is a bully?
**Q10.** What is the probability that a student in this class is both a bully and a victim?
**Q11.** What is the probability that a student in this class has not been involved in the cyber-bullying incidents in any way?

**Answers:**

**A1.** B = { Aziz, Bob, Dave, Ed, Frog }

**A2.** V = { Bob, Crazee, Gene, Ike }

**A3.** B ∩ V = { Bob }

**A4.** B ∩ T = { Bob, Frog }

**A5.** V ∩ T = { Bob, Gene }

**A6.** None. B ∩ V ∩ T = ∅

**A7.** Kerry is neither a bully nor a victim. Perhaps Kerry is a sensible by-stander who reported the bullying of other students to the teacher.

**A8.** (B ∪ V ∪ T) = { Jack, Lily }

**A9.** Probability of being a bully = ^{5}⁄_{12}

**A10.** Probability of being a bully and a victim = ^{1}⁄_{12}

**A11.** Probability of not being involved = ^{2}⁄_{12} = ^{1}⁄_{6}

## Question - Party Food

Draw a Venn diagram for the following data.

- 10 people at a party are Al, Bert, Cuc, David, Emon, Freda, Giuseppe, Henri, Ines and Jock.
- Al, Bert and Jock like hamburgers.
- Cuc, David, Emon, Giuseppe, Ines and Jock like pizza.
- Cuc, Ines and Jock like noodles.
- Freda and Henri are vegetarians who do not eat any of the party food.

## Maths Fun - Olympic Rings

Write **data about sports** for a Venn Diagram in the shape of the five Olympic Rings.