Symmetric Difference Calculator

What is a Symmetric Difference?

The symmetric difference between two sets is the set of elements that are in either of the sets, but not in both. It's like finding what's unique to each set! For sets A and B, we write this as A △ B or A ⊖ B.

How to Calculate Symmetric Difference

To find the symmetric difference between two sets, we need to:

1. Find elements that are in Set A but not in Set B
2. Find elements that are in Set B but not in Set A
3. Combine these unique elements to get the symmetric difference

Formula

The symmetric difference can be written in set notation as:

\[ A \triangle B = (A \setminus B) \cup (B \setminus A) \]

Or alternatively:

\[ A \triangle B = (A \cup B) \setminus (A \cap B) \]

Where:

• \(A \triangle B\) is the symmetric difference
• \(A \setminus B\) means elements in A but not in B
• \(B \setminus A\) means elements in B but not in A
• \(\cup\) means union (combining sets)
• \(\cap\) means intersection (common elements)

Example

Let's look at an example:

Set A = {1, 2, 3, 4}

Set B = {3, 4, 5, 6}

To find A △ B:

• Elements in A but not in B: {1, 2}
• Elements in B but not in A: {5, 6}
• Therefore, A △ B = {1, 2, 5, 6}

Visual Representation

In a Venn diagram, the symmetric difference is shown by the areas that belong to only one set - the non-overlapping regions. It's like highlighting the "exclusive" parts of each set!