Model of Knowledge by Fitzgerald Systems

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====== Notation ======


Consider a Universal set of (U) = {1, 3, 5, 8, 10, 35, 36, 37}


  * {}; set; A collection of elements; (A) = {1, 3, 5, 8}, (B) = {5, 8, 10, 35}, (C) = {8, 10, 36, 37}
  * a ∈ A; element of; Element a is a member of set A; 3 ∈ {1, 3, 5, 8}
  * a ∉ A; negation of a ∈ A
  * ∀; for all; ∀ x, P(x) means P(x) is true for all of x
  * ∃; existential quantification; ∃ x : P(x) means there exists at least one x such that P(x) is true
  * ∃!; uniqueness quantification; ∃! x : P(x) means there exists exactly one x such that P(x) is true
  * A ∪ B; union; Elements that belong to A or B; A ∪ B = {1, 3, 5, 8, 10, 35}
  * A ∩ B; intersection; Elements that belong to A and B; A ∩ B = {5, 8}
  * A = B; equality; Elements of set A are the same as elements of set B; {1, 2} = {1, 2}
  * A ⊆ B; subset; Every element of A is also an element of B; {1, 3} ⊆ {1, 3, 5, 8, 10, 35}
  * A ⊂ B; proper subset; Every element of A is also an element of B but A ≠ B; {1, 3} ⊆ {1, 3, 5, 8, 10, 35}
  * A ⊄ B; negation of A ⊂ B
  * A ⊇ B; superset; Every element of B is also an element of A; {1, 3, 5, 8, 10, 35} ⊇ {1, 3}
  * A ⊃ B; proper superset; Every element of B is also an element of A but B ≠ A; {1, 3, 5, 8, 10, 35} ⊃ {1, 3}
  * A \ B or A - B; relative complement; Elements that are members to A but not members to B; A - B = {1, 3}
  * A' or A^c or A with bar over the top; complement; Every element in Universal set but not in A; A^c = {10, 35, 36, 37}
  * {} or Ø; null set or empty set; A set with no elements;
  * P(A); power set; Given a set A, power set of A is the set of all subsets of the set; P({a, b}) = {Ø, {a}, {b}, {a,b}}
  * A Δ B; symmetric difference; Element that are in A or B but not members of the intersection of A and B; A Δ B = {1, 3, 10, 35}
  * |A| or #A; cardinality; Number of elements of set A; |A| = 4
  * A x B; cartesian product; Set of all ordered pairs from A and B; {1,2} x {3,4} = {(1,3), (1,4), (2,3), (2,4)}
  * ℵ; natural numbers, positive whole numbers
  * Z; integers, negative or positive whole numbers and zero
  * R; real numbers; R = {x : -∞ < x < ∞}
  * →; to; f : X → Y means function f maps the set X into set Y
  * |→; maps to f :x |→ y means function f maps the element x to the element y

====== Set Builder Notation ======

A = {x : x ∈ ℵ} the set A is equal to a set of x such that (| can also be used) x are elements of N (Natural numbers)

====== Arithmetic ======

  * n(A ∪ B) = n(A) + n(B) – n(A ∩ B)
Model of Knowledge by Fitzgerald Systems

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