Set Theory

Sets

• set = collection of objects (called elements)
  • roster notation -> A =
  • order and multiplicity dont matter
• symbols
  • ∈ is used to indicate that an element is in a set, as in 2 ∈ A (2 is an element of A, 2 belongs to A, A contains 2)
  • ∉ indicates that an element is not in a set, as in 5 ∉ A
• The set with no elements is called the empty set/null set and is denoted by the symbol ∅ / { }.
• The cardinality of a finite set A, denoted by |A|, is the number of distinct elements in A.
  • If A = { 2, 4, 6, 10 }, then |A| = 4. The cardinality of the empty set |∅| is zero.
• ellipses (...) are used to denote a long (possibly infinite) sequence of numbers.
• Common Sets (integers etc)

• R+ = set of all positive real numbers

• Z- = set of all negative integers.

• R* = set of all nonzero real numbers

• subsets and supersets

  • If every element in A is also an element of B, then A is a subset of B, denoted as A ⊆ B. If there is an element of A that is not an element of B, then A is not a subset of B, denoted as A ⊈ B.
  • If A ⊆ B and there is an element of B that is not an element of A (i.e., A ≠ B), then A is a proper subset of B, denoted as A ⊂ B.

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• power sets

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  • 2^elements

• different operators

  • difference
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  • symmetric difference
    • A ⊕ B = ( A - B ) ∪ ( B - A )
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  • complement
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• set identities

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• disjoint

  • 2 sets whose intersection is empty = are disjoint

• pairwise disjoint

  • every pair of distinct sets in a sequence is disjoint

• artition

  • a collection of non-empty subsets of a set such that every element is in exactly one of the subsets
    • slicing the pie