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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.

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:

\[ A \oplus B  = \{ 5 , 7, 8 \} \]

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