Set operation

What are Set Operations?
In mathematics, a set is a collection of distinct objects or elements. Set operations are ways of combining or manipulating these sets to create new ones. Think of them like basic arithmetic operations (addition, subtraction, etc.) but for sets instead of numbers. These operations are fundamental to fields like computer science, logic, and probability.
The most common set operations are union, intersection, and difference. They are often visualized using Venn diagrams, which show the relationships between different sets with overlapping circles.
Here's a breakdown of the main operations:
1. Union (A \cup B)
The union of two sets, A and B, is a new set that contains all the elements from both A and B, without any duplicates.
* Description: It combines the two sets into one larger set.
* Example: If set A = \{1, 2, 3\} and set B = \{3, 4, 5\}, then the union A \cup B = \{1, 2, 3, 4, 5\}. Notice that the number 3 is only listed once.
2. Intersection (A \cap B)
The intersection of two sets, A and B, is a new set that contains only the elements that are common to both A and B.
* Description: It finds the "overlap" between the two sets.
* Example: If set A = \{1, 2, 3\} and set B = \{3, 4, 5\}, then the intersection A \cap B = \{3\}.
3. Difference (A - B or A \setminus B)
The difference of two sets, A and B, is a new set containing all the elements that are in set A but are not in set B.
* Description: It removes the elements of B from A. The order matters here: A - B is not the same as B - A.
* Example: If set A = \{1, 2, 3\} and set B = \{3, 4, 5\}, then the difference A - B = \{1, 2\}.
* In contrast, B - A = \{4, 5\}.
4. Complement (A^c or A')
The complement of a set A is a new set that contains all the elements that are not in A, but are within a defined universal set (U).
* Description: It's everything outside of A, but still within the larger context or "universe" you're working in.
* Example: If the universal set U = \{1, 2, 3, 4, 5\} and set A = \{1, 2\}, then the complement A^c = \{3, 4, 5\}.