꒰ᐢ. .ᐢ꒱₊˚⊹

Sets

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Sets

  • An unordered collection of distinct objects
    • Duplicate elements does not change the set
  • Elements/members are objects in a set
    • 𝑎∈𝐴 denotes that 𝑎 is an element of the set 𝐴
    • 𝑎∉𝐴 denotes that 𝑎 is not an element of set 𝐴
  • Must be well-defined; membership can be verified with a yes/no answer

Set Examples

  • 𝑆={1,car,cat,c++,2.03}
  • 𝐵={1,3,{2},S}
  • 𝑌={𝑥|𝑥is a positive integer less than5}

Common Sets

  • 𝕎={𝑥|𝑥is a positive integer or zero}
  • ℕ={𝑥|𝑥is a positive integer}
  • ℤ={𝑥|𝑥is an integer}
  • ℤ+={𝑥|𝑥is a positive integer}
  • ℚ={𝑥|𝑥is a rational number}
  • ℝ={𝑥|𝑥is a real number}

Universal Set

  • A set that contains everything within a certain context
  • Typically represented with a rectangle in diagrams, denoted with 𝑈 or 𝜉

Empty Set

  • A set containing no elements is denoted with ⌀ or {}

Subsets

  • The set 𝐴 is a subset of 𝐵 if every element of 𝐴 is also an element of 𝐵
  • The notation 𝐴⊆𝐵 denotes that 𝐴 is a subset of 𝐵
    • {1,2}⊆{1,2,3}
    • 1⊄{1,2,3} but {1}⊆{1,2,3}

Proper Subsets

  • If every element of 𝐴 is an element of 𝐵, but not every element in set 𝐵 is in set 𝐴
    • 𝐴 must be shorter than 𝐵
  • Denoted with 𝐴⊂𝐵, rather than 𝐴⊆𝐵 since 𝐴≠𝐵

Supersets

  • If every element of 𝐵 is an element of 𝐴, then 𝐴 is a subset of 𝐵
  • Denoted with 𝐴⊇𝐵

Cardinality

  • The number of distinct elements in a set
  • Denoted with |𝑆|
    • |{7,7,7,7}|=1

Inclusion-Exclusion Principle

  • |𝐴∪𝐵|=|𝐴|+|𝐵|−|𝐴∪𝐵|

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