Set Theory
What Are Sets?
Sets are collections of distinct objects. These objects can be numbers, letters, or even other sets! Sets help us organize and classify objects by grouping them together.
Notation and Operations
Sets are typically written with curly braces, like \( S = {1, 2, 3} \). You can perform operations with sets:
- Union (\(A \cup B\)): Combines all elements from both sets.
- Intersection (\(A \cap B\)): Finds elements common to both sets.
- Difference (\(A - B\)): Finds elements in one set but not the other.
- Subset (\(A \subseteq B\)): Checks if all elements in \(A\) are also in \(B\).
Why Are Sets Important?
Sets are the foundation of most mathematical concepts in discrete math. They help us describe and analyze groups of objects in logic, computer science, and more.
Real-World Connections
Collections you use every day can be thought of as sets, like your group of friends, the books you own, or the flavors of ice cream at a shop!
Examples
The set of vowels in the alphabet: \({a, e, i, o, u}\)
The union of \({1, 2, 3}\) and \({3, 4, 5}\) is \({1, 2, 3, 4, 5}\)
In a Nutshell
Sets are collections of distinct objects, and we use them to organize and compare groups.