A **set **is a collection of items.

We denote a set using a capital letter and we define the items within the set using curly brackets. For example, suppose we have some set called “A” with elements 1, 2, 3. We would write this as:

**A = {1, 2, 3}**

This tutorial explains the most common **set operations **used in probability and statistics.

**Union**

**Definition: **The *union *of sets A and B is the set of items that are in either A or B.

**Notation: **A ∪ B

**Examples:**

- {1, 2, 3} ∪ {4, 5, 6} = {1, 2, 3, 4, 5, 6}
- {1, 2} ∪ {1, 2} = {1, 2}
- {1, 2, 3} ∪ {3, 4} = {1, 2, 3, 4}

**Intersection**

**Definition: **The *intersection *of sets A and B is the set of items that are in both A and B.

**Notation: **A ∩ B

**Examples:**

- {1, 2, 3} ∩ {4, 5, 6} = {∅}
- {1, 2} ∩ {1, 2} = {1, 2}
- {1, 2, 3} ∩ {3, 4} = {3}

**Complement**

**Definition: **The *complement *of set A is the set of items that are in the universal set U but are not in A.

**Notation: **A’ or A^{c}

**Examples:**

- If U = {1, 2, 3, 4, 5, 6} and A = {1, 2}, then A
^{c}= {3, 4, 5, 6} - If U = {1, 2, 3} and A = {1, 2}, then A
^{c}= {3}

**Difference**

**Definition: **The *difference *of sets A and B is the set of items that are in A but not B.

**Notation: **A – B

**Examples:**

- {1, 2, 3} – {2, 3, 4} = {1}
- {1, 2} – {1, 2} = {∅}
- {1, 2, 3} – {4, 5} = {1, 2, 3}

**Symmetric Difference**

**Definition: **The *symmetric difference *of sets A and B is the set of items that are in either A or B, but not in both.

**Notation: **A Δ B

**Examples:**

- {1, 2, 3} Δ {2, 3, 4} = {1, 4}
- {1, 2} Δ {1, 2} = {∅}
- {1, 2, 3} Δ {4, 5} = {1, 2, 3, 4, 5}

**Cartesian Product**

**Definition: **The *cartesian product *of sets A and B is the set of ordered pairs from A and B.

**Notation: **A x B

**Examples:**

- If A = {H, T} and B = {1, 2, 3}, then A x B = {(H, 1), (H, 2), (H, 3), (T, 1), (T, 2), (T, 3)}
- If A = {T, H} and B = {1, 2, 3}, then A x B = {(T, 1), (T, 2), (T, 3), (H, 1), (H, 2), (H, 3)}