After understanding union and intersection, the next important operations in set theory are:
These operations help identify elements that are exclusive to one set or differ between sets.
The difference of two sets A and B is the set of elements that are in A but not in B.
A − B = { x | x ∈ A and x ∉ B }
ABA = {1, 2, 3, 4}
B = {3, 4, 5}
A − B = {1, 2}
✔ Elements common with B are removed
B − A = {5}
⚠ Important:
A − B ≠ B − A
A − A = ∅
A − ∅ = A
A − U = ∅
A − B = A ∩ B'
The symmetric difference of sets A and B is the set of elements that are in A or B but not in both.
A Δ B
A Δ B = (A − B) ∪ (B − A)
A = {1, 2, 3}
B = {3, 4, 5}
Step 1:
A − B = {1, 2}
B − A = {4, 5}
Step 2:
A Δ B = {1, 2, 4, 5}
A Δ B = B Δ A
(A Δ B) Δ C = A Δ (B Δ C)
A Δ ∅ = A
A Δ A = ∅
| Feature | Set Difference (A − B) |
Symmetric Difference (A Δ B) |
|---|---|---|
| Meaning | Elements in A only | Elements in A or B but not both |
| Symmetry | Not symmetric | Symmetric |
| Example | {1,2} |
{1,2,4,5} |
Set Difference (A − B): Elements in A but not in B
Symmetric Difference (A Δ B): Elements in A or B but not both
Important relations:
A − B = A ∩ B'A Δ B = (A − B) ∪ (B − A)