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