Intuitive Set Theory

Intuitive Set Theory
Some Notation

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: $U$

There's Always More to Learn