1. What Is the Intersection of Sets?
The intersection of sets contains only those elements that are common to both sets. It focuses on the shared part of the sets.
If an element is present in both A and B, then it is part of their intersection.
2. Notation
The symbol for intersection is:
\( A \cap B \)
This is read as “A intersection B”.
3. Meaning of A ∩ B
\( A \cap B \) contains:
- Only the elements that A and B share
- No elements that are in just one set
4. Examples
Here are some clear examples:
4.1. Example 1
\( A = \{1,2,3\}, B = \{3,4,5\} \Rightarrow A \cap B = \{3\} \)
4.2. Example 2
\( X = \{a,b,c\}, Y = \{b,c,d\} \Rightarrow X \cap Y = \{b,c\} \)
4.3. Example 3 (No Intersection)
If the sets have no common elements:
\( P = \{1,2\}, Q = \{3,4\} \Rightarrow P \cap Q = \emptyset \)
5. Intersection of More Than Two Sets
The idea extends to three or more sets. It means elements that are common to all sets.
5.1. Example
A = \{1,2,3\}, B = \{2,3,4\}, C = \{3,5\}
Then:
\( A \cap B \cap C = \{3\} \)
6. Intersection in Venn Diagrams
In a Venn diagram, the intersection is shown by shading the overlapping region of the circles.
7. Important Points
A few ideas to remember about intersections:
7.1. Key Ideas
- Only common elements are included.
- If sets do not share elements, their intersection is empty.
- Intersection works the same for any number of sets.
- Order does not matter: \( A \cap B = B \cap A \).