 # Find the symmetric difference of {1, 3, 5} and {1, 2, 3}. 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.

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:

$A \oplus B = \{ 5 , 7, 8 \}$