This **article aims to find the symmetric difference between two sets**. The article uses the **definition of symmetric difference. **Suppose there are **two sets**, **A** and **B**. The **symmetric difference** between the two sets **A** and **B** is the set that **contains the elements present** in both sets except the **common elements**.

A **symmetric difference** between two sets is also called **disjunctive conjunction**. A **symmetric difference** between two sets is the** set of elements** that are in both sets but not in their **intersection**.

**Expert Answer**

**Given **

\[ A = \{ 1 , 3 , 5 \} \]

\[ B = \{ 1 , 2 , 3 \} \]

We notice that $ 1 $ and $ 3 $ **are in both sets**. So $ 1 $ and $ 3 $ are $ NOT $ in **symmetric difference**

\[ A \oplus B \]

$ 5 $ is an** element** of **A** that is **not** in **B**. So $ 5 $ is in the **symmetric difference** $ A \oplus B $.

\[ 5 \in A \oplus B \]

$2$ is an** element** of **A** that is **not** in **B**. So $ 2 $ is in the **symmetric difference** $ A \oplus B $.

\[ 2 \in A \oplus B \]

Then we’ve gone through **all the elements** in **A** and **B**, so the only elements in **symmetric difference** $ A \oplus B $ are then $ 2 $ and $ 5 $:

\[ A \oplus B = \{ 2 , 5 \} \]

**Numerical Result**

The **symmetric difference** is given as:

\[ A \oplus B = \{ 2 , 5 \} \]

**Example**

**Find the symmetric difference of { 1 , 2 , 3 , 5 , 7 } and { 1, 2, 3, 8 }.**

**Solution**

**Given **

\[ A = \{ 1, 2 , 3, 5, 7 \} \]

\[ B = \{ 1 , 2 , 3 , 8 \} \]

We notice that $ 1 $, $ 2 $ and $ 3 $ **are in both sets**. So $ 1 $, $ 2 $ and $ 3 $ are **NOT** in **symmetric difference**

\[ A \oplus B \]

$ 5 $ is an** element** of **A** that is **not** in **B**. So $ 5 $ is in the **symmetric difference** $ A \oplus B $.

\[ 5 \in A \oplus B \]

$ 7 $ is an** element** of **A** that is **not** in **B**. So $ 7 $ is in the **symmetric difference** $ A \oplus B $.

\[ 7 \in A \oplus B\]

$ 8 $ is an** element** of **B** that is **not** in **A.** So $ 8 $ is in the **symmetric difference** $ A \oplus B $.

\[ 8 \in A\oplus B \]

Then we’ve gone through **all the elements** in **A** and **B**, so the only elements in **symmetric difference** $ A \oplus B $ are then $ 5 $ , $ 7 $ and $ 8 $:

\[ A \oplus B = \{ 5 , 7 , 8 \} \]

The **symmetric difference** is given as:

\[ A \oplus B = \{ 5 , 7, 8 \} \]