Intuitive Set Theory

Intuitive Set Theory
Some Notation

Prof. David Bernstein
James Madison University

Computer Science Department

bernstdh@jmu.edu

Sets

Sets:
- A set is a collection of elements
- The number of elements in a set is its cardinality
Defining Sets:
- Extensive Definition - List all of the elements in the set
  \(A = \{ a, b, g, m \}\)
- Intensive Definition - Use the properties of the elements
  \(B = \{ x: x \in \mathbb{R}, x \leq 100 \}\)

Relationships

Relations between Entities and Sets:
- \(x \in A\) indicates the entity/element \(x\) is a member of the set \(A\)
- \(x \not\in A\) indicates the entity/element \(x\) is not a member of the set \(A\)
Relations between Sets:
- Equality: \(A = B\) indicates \(A\) and \(B\) have the same elements
- Inequality: \(A \neq B\) indicates \(A\) and \(B\) do not have the same elements
- Inclusion: \(A \subseteq B\) indicates every element of \(A\) is also an element of \(B\) (i.e., if \(x \in A\) then \(x \in B\))
- Subset: \(A \subset B\) indicates \(A \subseteq B\) and \(A \neq B\)

Operations and Special Sets

Operations:
- Union: \(A \cup B = \{x : x \in A \text{ or } x \in B \}\)
- Intersection: \(A \cap B = \{x : x \in A \text{ and } x \in B \}\)
- Absolute Complement: \(\overline{A} = \{x : x \not\in A \}\)
- Relative Complement: \(B - A = \{x : x \in B \text{ and } x \not\in B \}\)
Special Sets:
- Empty Set: \(\emptyset\)
- Universal Set: \(U\)