Set Theory Summary

Sets Definition

• Set: A well-defined collection of objects.
• Elements: The objects that are part of a set.
• Notation:
  • Sets are denoted with upper-case letters.
  • Elements are denoted with lower-case letters.
  • Membership: x ∈ A (x is a member of A), x ∉ A (x is not a member of A).

Ways of Describing Sets

• List Elements: Directly list the members.
• Verbal Description: E.g., "A is the set of all integers from 1 to 6, inclusive."
• Mathematical Inclusion Rule: Define elements through a specific rule.

Special Sets

• Null Set (Empty Set): Symbolized by ∅, contains no elements.
• Universal Set: Contains all elements currently under consideration, symbolized by U.

Membership Relationships

• Subset: A is a subset of B if all elements of A are also in B. Notation: A ⊆ B.
• Proper Subset: A is a proper subset of B if A is a subset of B and A is not equal to B. Notation: A ⊂ B.

Combining Sets

• Set Union (A ∪ B): Set of all elements in A, in B, or in both.
• Set Intersection (A ∩ B): Set of all elements in both A and B.
• Set Complement (¬A): Set of all elements not in A.
• Set Difference (A - B): Set of all elements in A with elements of B removed.

Examples

• If A = {1,2,3} and B = {3,4,5,6}, then:
  • A ∩ B = {3}
  • A ∪ B = {1,2,3,4,5,6}
  • B - A = {4,5,6}

Venn Diagrams

• Visual representation of sets and their relationships.

Mutually Exclusive and Exhaustive Sets

• Exhaustive Sets: Their union equals the entire set.
• Mutually Exclusive Sets: No elements in common.

Set Partition

• Partitioning a set involves dividing it into mutually exclusive and exhaustive subsets.

Test Questions

• A ∪ Ø = ?
• A ∩ A = ?
• A - A = ?
• A ∩ Ø = ?
• If A ⊆ B then A ∩ B = ?