Chapter: Sets

• Introduction & Definition

  • Well-defined collection of objects
  • Objects are called elements or members
  • Usually denoted by capital letters ($A, B, C$)
  • Elements denoted by lowercase letters ($a, b, c$)
  • Membership notation: $a \in A$ (belongs to), $b \notin A$ (does not belong to)

• Representation of Sets

  • Roster Form (Tabular Form)
    • List all elements separated by commas inside braces { }
    • Example: $A = {2, 4, 6, 8}$
  • Set-Builder Form
    • State a characterizing property
    • Example: $A = {x : x \text{ is an even natural number less than } 10}$

• Types of Sets

  • Empty Set (Null / Void Set)
    • Contains no elements
    • Denoted by $\phi$ or ${}$
  • Finite Set
    • Countable number of elements
  • Infinite Set
    • Infinite number of elements
  • Equal Sets
    • Exactly the same elements ($A = B$)

• Subsets & Power Set

  • Subset ($A \subseteq B$)
    • Every element of $A$ is also in $B$
  • Proper Subset ($A \subset B$)
    • $A \subseteq B$ and $A \neq B$
  • Universal Set ($U$)
    • Contains all possible elements under consideration
  • Power Set
    • Set of all subsets of a set
    • Total subsets of set with $n$ elements = $2^n$

• Intervals as Subsets of $\mathbb{R}$

  • Open Interval: $(a, b) = {x : a < x < b}$
  • Closed Interval: $[a, b] = {x : a \le x \le b}$
  • Semi-open / Semi-closed: $[a, b)$ or $(a, b]$

• Venn Diagrams

  • Rectangular box represents Universal Set
  • Closed curves (circles) represent Subsets

• Operations on Sets

  • Union ($A \cup B$)
    • ${x : x \in A \text{ or } x \in B}$
  • Intersection ($A \cap B$)
    • ${x : x \in A \text{ and } x \in B}$
  • Difference ($A - B$)
    • Elements in $A$ but not in $B$
  • Complement ($A'$ or $A^c$)
    • $U - A$ (elements of $U$ not in $A$)

• Practical Problems (Set Theory)

  • Cardinality Formula
    • $n(A \cup B) = n(A) + n(B) - n(A \cap B)$
    • For three sets: $n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(A \cap C) + n(A \cap B \cap C)$