1. Union of Sets (A ∪ B)
The Union of two sets A and B is the set of all elements that are in A, or in B, or in both.
Notation: A ∪ B.
Example: Let A = {1, 2, 3} and B = {3, 4, 5}.
The union A ∪ B = {1, 2, 3, 4, 5}.
2. Intersection of Sets (A ∩ B)
The Intersection of two sets A and B is the set of all elements that are in both A and B.
Notation: A ∩ B.
Example: Let A = {1, 2, 3} and B = {3, 4, 5}.
The intersection A ∩ B = {3}.
3. Difference of Sets (A − B)
The Difference of two sets A and B (also called the relative complement of B in A) is the set of elements that are in A but not in B.
Notation: A − B.
Example: Let A = {1, 2, 3} and B = {3, 4, 5}.
The difference A − B = {1, 2}.
4. Complement of a Set (A' or A̅)
The Complement of a set A, denoted by A' or A̅, is the set of all elements in the universal set U that are not in A.
Notation: A' or A̅.
Example: If the universal set U = {1, 2, 3, 4, 5} and A = {1, 2}, then the complement A' = {3, 4, 5}.
5. Symmetric Difference (A Δ B)
The Symmetric Difference of two sets A and B is the set of elements that are in either of the sets A or B but not in their intersection.
Notation: A Δ B.
Example: Let A = {1, 2, 3} and B = {3, 4, 5}.
The symmetric difference A Δ B = {1, 2, 4, 5}.
6. Cartesian Product of Sets (A × B)
The Cartesian Product of two sets A and B is the set of all ordered pairs (a, b) where a is an element of A and b is an element of B.
Notation: A × B.
Example: Let A = {1, 2} and B = {x, y}.
The Cartesian product A × B = {(1, x), (1, y), (2, x), (2, y)}.
7. Important Points to Remember
- The Union of sets combines all elements from the involved sets.
- The Intersection identifies common elements between the sets.
- The Difference of sets highlights elements in one set but not in the other.
- The Complement represents everything outside a particular set within the universal set.
- The Symmetric Difference highlights elements that are unique to each set, excluding the intersection.
- The Cartesian Product results in pairs formed from elements of two sets.