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Intuitive Set Theory
Some Notation


Prof. David Bernstein
James Madison University

Computer Science Department
bernstdh@jmu.edu

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Sets
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  • 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
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  • 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
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  • 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
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