ðŸŠī Notes

Sets

2 min read

Course: CSCI 1900

Sets

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

Set Examples

  • 𝑆={1,car,cat,c++,2.03}
  • ðĩ={1,3,{2},𝑆}
  • 𝑌={ð‘Ĩ|ð‘Ĩis a positive integer less than5}

Common Sets

  • 𝕎={ð‘Ĩ|ð‘Ĩis a positive integer or zero}
  • ℕ={ð‘Ĩ|ð‘Ĩis a positive integer}
  • â„Ī={ð‘Ĩ|ð‘Ĩis an integer}
  • â„Ī+={ð‘Ĩ|ð‘Ĩis a positive integer}
  • ℚ={ð‘Ĩ|ð‘Ĩis a rational number}
  • ℝ={ð‘Ĩ|ð‘Ĩis 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 ðī is a subset of ðĩ if every element of ðī is also an element of ðĩ
  • The notation ðī⊆ðĩ denotes that ðī is a subset of ðĩ
    • {1,2}⊆{1,2,3}
    • 1⊄{1,2,3} but {1}⊆{1,2,3}

Proper Subsets

  • If every element of ðī is an element of ðĩ, but not every element in set ðĩ is in set ðī
    • ðī must be shorter than ðĩ
  • Denoted with ðī⊂ðĩ, rather than ðī⊆ðĩ since ðī≠ðĩ

Supersets

  • If every element of ðĩ is an element of ðī, then ðī is a subset of ðĩ
  • Denoted with ðī⊇ðĩ

Cardinality

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

Inclusion-Exclusion Principle

  • |ðī∊ðĩ|=|ðī|+|ðĩ|−|ðī∊ðĩ|

Graph View

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