1. What Are Set Operations?
Set operations are rules that help combine sets, compare them, or find parts of them. These operations allow us to work with sets in a simple and logical way, similar to how arithmetic operations work with numbers.
The most common set operations are union, intersection, difference, and complement.
2. Union of Sets
The union of two sets includes all elements that are in either of the sets or in both. It combines the sets together.
2.1. Notation
\( A \cup B \)
2.2. Meaning
All elements of A, all elements of B, without repeating anything.
2.3. Example
\( A = \{1,2\}, B = \{2,3\} \Rightarrow A \cup B = \{1,2,3\} \)
3. Intersection of Sets
The intersection of two sets contains only the elements that are common to both sets.
3.1. Notation
\( A \cap B \)
3.2. Meaning
The overlap between A and B.
3.3. Example
\( A = \{1,2,3\}, B = \{3,4\} \Rightarrow A \cap B = \{3\} \)
4. Difference of Sets
The difference of two sets shows the elements that are in one set but not in the other.
4.1. Notation
\( A - B \)
4.2. Meaning
Elements in A that are not in B.
4.3. Example
\( A = \{1,2,3\}, B = \{2,4\} \Rightarrow A - B = \{1,3\} \)
5. Complement of a Set
The complement of a set A includes everything in the universal set U that is not in A.
5.1. Notation
\( A' \)
5.2. Example
If U = {1,2,3,4,5} and A = {2,3}, then:
\( A' = \{1,4,5\} \)
6. Symmetric Difference
The symmetric difference of two sets includes elements that are in either A or B but not in both.
6.1. Notation
\( A \triangle B \)
6.2. Example
\( A = \{1,2,3\}, B = \{3,4,5\} \Rightarrow A \triangle B = \{1,2,4,5\} \)
7. Using Venn Diagrams for Set Operations
Venn diagrams give a clear visual way to understand how these operations work. The overlap, union, and exclusive regions can all be shown through shading.
7.1. Common Visuals
- Union → both circles shaded
- Intersection → overlapping area shaded
- Difference → one side of a circle shaded
- Complement → area outside the circle shaded
8. Important Points
A few simple ideas to keep in mind about set operations:
8.1. Key Ideas
- Union combines sets.
- Intersection finds common elements.
- Difference removes elements of one set from another.
- Complement depends on the universal set.
- Operations work the same even with more than two sets.