Question

Symmetric different of set

Answer 1

The symmetric difference of two sets, often denoted as \( A \triangle B \) or \( A \Delta B \), is defined as the set of elements that are in either of the sets \( A \) or \( B \), but not in their intersection. In other words, it contains those elements that are unique to each set.

Mathematically, the symmetric difference can be expressed using set notation as:

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

Where:
- \( A \setminus B \) is the set of elements that are in \( A \) but not in \( B \).
- \( B \setminus A \) is the set of elements that are in \( B \) but not in \( A \).
- \( \cup \) represents the union of two sets.

### Example

If we have the following two sets:
- \( A = \{1, 2, 3, 4\} \)
- \( B = \{3, 4, 5, 6\} \)

1. The elements in \( A \) that are not in \( B \) (i.e., \( A \setminus B \)):
\[
A \setminus B = \{1, 2\}
\]

2. The elements in \( B \) that are not in \( A \) (i.e., \( B \setminus A \)):
\[
B \setminus A = \{5, 6\}
\]

3. Now, we take the union of these two results:
\[
A \triangle B = \{1, 2\} \cup \{5, 6\} = \{1, 2, 5, 6\}
\]

Thus, the symmetric difference \( A \triangle B = \{1, 2, 5, 6\} \).

### Properties of Symmetric Difference
1. **Commutative**: \( A \triangle B = B \triangle A \)
2. **Associative**: \( A \triangle (B \triangle C) = (A \triangle B) \triangle C \)
3. **Identity**: \( A \triangle \emptyset = A \)
4. **Involution**: \( A \triangle A = \emptyset \)

These properties make the symmetric difference a useful and interesting operation in set theory.