1. What Is the Symmetric Difference?
The symmetric difference of two sets contains elements that are in either of the sets, but not in both. In simple words, it keeps the uncommon elements and removes the common ones.
It is like taking the union of the sets and then removing the intersection.
2. Notation
The symbol for symmetric difference is:
\( A \triangle B \)
This is read as “A symmetric difference B”.
3. Meaning of A △ B
\( A \triangle B \) contains:
- Elements in A but not in B
- Elements in B but not in A
- No elements that are in both
4. Examples
Here are clear examples to understand the symmetric difference:
4.1. Example 1
\( A = \{1,2,3\}, B = \{3,4,5\} \)
Common element is 3, so remove it. What is left:
\( A \triangle B = \{1,2,4,5\} \)
4.2. Example 2
\( X = \{a,b,c\}, Y = \{b,d\} \)
Common = {b}. Remove b from both:
\( X \triangle Y = \{a,c,d\} \)
4.3. Example 3 (No Intersection)
If sets do not overlap:
\{1,2\} \triangle \{3,4\} = \{1,2,3,4\}
because there are no common elements to remove.
5. Relationship With Union and Intersection
The symmetric difference can also be written using union and intersection:
5.1. Formula
\( A \triangle B = (A \cup B) - (A \cap B) \)
This means: take everything from A and B, then remove the common part.
6. Symmetric Difference With More Than Two Sets
The idea can continue for more sets, but the interpretation becomes more complex. For three sets A, B, C:
6.1. Concept
Elements appear in the result only if they occur in an odd number of sets.
6.2. Example
If A = {1,2}, B = {2,3}, C = {3,4}
The symmetric difference is:
(A △ B) △ C = \{1,3,4\}
7. Symmetric Difference in Venn Diagrams
In a Venn diagram, the symmetric difference of A and B is shown by shading both circles except the overlapping region.
8. Important Points
A few ideas to remember:
8.1. Key Ideas
- Symmetric difference keeps only the elements that are different between sets.
- Common elements are always removed.
- Order does not matter: \( A \triangle B = B \triangle A \).
- Useful for comparing differences between groups.