What is a Set?
A set is a well-defined collection of distinct objects. These objects are called elements or members of the set. Sets are usually written using curly braces.
Example:
\(A = \{ 1,2,3,4,5\}\)
Here, \(3 \in A\) (3 is an element of \(A\)) and \(7 \notin A\) (7 is not an element of \(A\)).
Common Set Notation
| Symbol | Meaning |
|---|---|
| \(\in\) | Element of |
| \(\notin\) | Not an element of |
| \(⊆\) | Subset of |
| \(∪\) | Union |
| \(∩\) | Intersection |
| \(A'\) | Complement of A |
| \(∅\) | Empty Set |
Set Operations
Let:
\(A = \{ 1,2,3,4\}\)
\(B = \{ 3,4,5,6\}\)
Union \((A \cup B)\)
All elements found in either set.
\(\{ 1,2,3,4,5,6\} \)
Intersection \((A \cap B)\)
Elements common to both sets.
\(\{ 3,4\} \)
Difference \((A - B)\)
Elements in \(A\) but not in \(B\).
\(\{ 1,2\} \)
Venn Diagram Concept
Venn diagrams visually represent relationships between sets. Overlapping regions show intersections, while the entire combined area represents the union of the sets.
Practice Problems
Let \(A = \{ 1,2,3,4\} \) and \(B = \{ 3,4,5,6\} \). Find \(A \cup B\).
Let \(A = \{ 2,4,6,8\} \) and \(B = \{ 1,2,3,4\} \). Find \(A \cap B\).
Determine whether \(\{ 1,2\} \subseteq \{ 1,2,3,4\} \).
Let \(A = \{ a,b,c,d \} \) and \(B = \{ c,d,e \} \). Find \(A - B\).
How many elements are in the set \(\{ 5, 10, 15, 20, 25 \} \)?
Challenge Problems
-
Let \(A = \{ 1,2,3,4,5 \} \) and \(B = \{ 4,5,6,7 \} \). Find:
- \(A \cup B\)
- \(A \cap B\)
- \(B - A\)
- Is the empty set a subset of every set? Explain.
- Create two sets whose intersection contains exactly three elements.