set_theory.theory.txt

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                                  ┃   SET_THEORY   ┃
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Collection:
  - bunch of elements

Class:
  - collection which can be defined

Universe:
  - noted U
  - class containing all possible sets for a specific object of study

Family:
  - function using classes|sets as domain|codomain

Set:
  - notation: {[VAL,...]}
  - unordered, i.e. position is insignificant
  - duplicates are ignored

Tuple:
  - also named vector
  - notation: ([VAL,...])
  - ordered
  - can contain duplicates
  - n-tuple:
     - tuple with n elements

Cardinality:
  - also named size
  - notation:
     - |A|
     - #A
     - A̿
     - n(A)
     - card(A)
  - number of elements in set

Empty set:
  - notation: {} or ∅
  - set with cardinality 0

Mapping:
  - notation: {MAP:f(VAR,...)}
     - VAR can be used in MAP
  - set with:
     - all elements where f(VAR,...) is true
     - elements mapped with MAP

Enumeration:
  - notation: {x, ..., y}
  - ... are all elements from x to y

Intervals:
  - set of all elements between a minimum and a maximum value
  - closed:
     - noted [a,b]
     - {x: a <= x <= b}
  - open:
     - noted (a,b) or ]a,b[
     - {x: a < x < b}
  - semi-open:
     - noted (a,b], [a,b)
     - mix of above
  - proper: when neither:
     - degenerate: when a = b
     - empty: when max < min
  - [left|right-]unbounded:
     - when a or b = Inf
     - inverse: bounded|finite

Set membership:
  - notation:
     - VAL,...∈ SET
     - VAL,...∉ SET (inverse)
  - means VAL are all [not] members of SET

Subset|superset:
  - operations
     - A ⊂ B: proper subset (not equal)
     - A ⊃ B: proper superset (not equal)
     - A ⊆ B: subset or equal
     - A ⊇ B: superset or equal
  - formulas:
     - A ⊂ B  => A ⊆ B
     - A ⊂ B <=> B ⊃ A
     - A ⊆ B <=> B ⊇ A
     - A ⊂ B <=> A ⊉ B
     - B ⊃ A <=> B ⊈ A

Complement:
  - notations:
     - ~A
     - A^c (superscript c)
  - means {x:x∉ A}

Basic combinations:
  - types:
     - intersection:
        - noted A ∩ B or A & B
        - {x: (x∈ A) && (x∈ B)}
     - union:
        - noted A ∪ B or A | B
        - {x: (x∈ A) || (x∈ B)}
  - commutative
  - associative
  - distributivity:
     - (A & C) | (B & C) = (A | B) & C
     - (A | C) & (B | C) = (A & B) | C
  - visualized with Venn diagrams:
     - each set is a circle
     - their area represents their elements
     - colors show result of operations

Advanced combinations
  - difference:
     - also named relative complement
     - noted A \ B or A - B
     - same as A & ~B
  - symmetric difference:
     - noted A ⊕ B or A xor B
     - same as (A | B) \ (A & B)

Mutual exclusiveless
  - also named disjoints
  - when A & B = ∅

Cartesian product:
  - noted A * B * ...
  - {(a,b,...): a∈ A, b∈ B,...}
  - formulas:
     - |A * B| = |A| * |B|

Power set:
  - noted P(A) or 2^A (superscript)
  - set with all possible subsets of a set
  - formulas:
     - |P(A)| = 2 ** |A|

Kleene star:
  - also named Kleene operator|closure
  - noted A*
  - tuple of any size (including 0) where each element∈ A

Kleene plus:
  - noted A⁺
  - tuple of any size (excluding 0) where each element∈ A