Venn diagrams are visual representations of groups that we call **sets**. Set notation or Venn diagram notation are the mathematical symbols that describe the different parts of a Venn diagram. There are over 30 different symbols used with Venn diagrams, but you only need to know a small section of these.

### John Venn

John Venn (1834 – 1923) was an English mathematician, logician and philosopher. He’s best known for introducing Venn diagrams. These diagrams are named after him, so Venn is always written with a capital letter.

### How Are Venn Diagrams Used?

Venn diagrams can be used to show different **sets** of items. The things in a set are called the **elements**.

For example, you could use a Venn diagram to show the multiples of 3 and the factors of 60. In this case, one of the sets (the multiples of 3) would be **infinite** so we might limit the numbers we are considering – we might only use the integers that are less than or equal to 30.

The set of numbers we are considering is called the **universal set** and is denoted by the Greek letter xi: ξ. The elements in the universal set will vary; they could be all the numbers under 100, any polygon with less than 10 sides or all the countries in the world.

The Venn diagram below shows all the integers less than or equal to 30. Set A is the factors of 60 and set B is the multiples of 3.

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### Intersects, Unions and Complements

The area where the two circles overlap is called the **intersect**. We denote it using the symbol ∩. In the Venn diagram above, the set A ∩ B = {3, 16, 12, 15, 30}. These are the integers that are **both** multiples of 3 and factors of 60.

The area that includes both circles is called the **union**. We denote it using the symbol ∪. In the Venn diagram above, the set A ∪ B = {1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 21, 24, 27, 30}. These are the integers that are multiples of 3 **or** factors of 60 **or both**.

The area outside a set is called the **complement**. We denote it using the symbol ‘. In the Venn diagram above, the set B’ = {1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26, 28, 29}. These are the integers that are **not **multiples of 3.

These three symbols are shown by the shaded sections below.

### Three Practice Questions Using Venn Diagram Notation

**1. **Given ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {2, 4, 6, 8, 10} and B = {2, 3, 6, 7, 8}, list the numbers in A ∪ B.

- a. {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
- b. {2, 3, 4, 6, 7, 8, 10}
- c. {2, 6, 8}
- d. {1, 5, 9}

**2**. Given ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {2, 4, 6, 8, 10} and B = {2, 3, 5, 6, 7, 8}, list the numbers in A ∩ B.

**a.**{1, 2, 3, 4, 5, 6, 7, 8, 9, 10}**b.**{2, 3, 4, 6, 7, 8, 10}**c.**{2, 6, 8}**d.**{1, 5, 9}

**3**. Given ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {2, 4, 6, 8, 10} and B = {2, 3, 5, 6, 7, 8}, list the numbers in A’.

**a.**{2, 4, 6, 8, 10}**b.**{1, 3, 5, 7, 9}**c.**{3, 5,}**d.**{1, 5, 9}

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