Introduction:
Set Theory involves sets which are unordered lists.
Ex. Set: S = (a, b, c)
Note: S is the set and a, b, c are the elements of the set.
- All members of a set must be of the same type. (Note: because set can have complex types (sets themselves); this can be circumvented by the properties of the complex types.)
Set Membership:
- ∅ element of
- ∅ = {}
- ∅ denotes the empty set
- ∈ element of
- Set: S = {a, b, c}
- a ∈ S
- ∉ not element of
- Set: S = {a, b, c}
- d ∉ S
- ⊂ proper (strict) subset of
- Set: S = {a, b, c}
- Set: T = {a, b}
- T ⊂ S
- ⊄ not proper (strict) subset of
- Set: S = {a, b, c}
- Set: Q = {a, b, c}
- Set: W = {b, c, d}
- Q ⊄ S
- W ⊄ S
- ⊆ subset of
- Set: S = {a, b, c}
- Set: Q = {a, b, c}
- Set: T = {a, b}
- Q ⊆ S
- T ⊆ S
- ⊈ not subset of
- Set: S = {a, b, c}
- Set: W = {b, c, d}
- W ⊄ S
- ⊃ superset of (strict)
- Set: S = {a, b, c}
- Set: T = {a, b}
- S ⊃ T
- ⊅ not superset of (strict)
- Set: S = {a, b, c}
- Set: Q = {a, b, c}
- Set: W = {b, c, d}
- Q ⊅ S
- W ⊅ S
- ⊇ superset of (not strict)
- Set: S = {a, b, c}
- Set: Q = {a, b, c}
- Set: T = {a, b}
- Q⊇S
- T⊇S
- ⊉ not superset of (not strict)
- Set: S = {a, b, c}
- Set: W = {b, c, d}
- W⊉S
Set Operators:
- ∩ disjoint
- Set: S = (a, b, c, d)
- Set: T = (a, b, e, f)
- S ∩ T = (c, d, e, f)
- ∪ union
- Set: S = (a, b, c, d)
- Set: T = (a, b, e, f)
- S ∪ T = (a, b)
- \ difference (relative complement)
- S = (a, b, c, d)
- Set: T = (a, b, e, f)
- S \ T = (c, d)
- Note: (c, d) is the relative complement
- T \ S = (e, f)
- Note: (c, d) is the relative complement
- absolute complement
- Set: S = (a, b, c, d, e, f)
- Set: T = (a, b)
- Set: W = (c, d)
- T ⊂ S
- T ⊂ W
- The absolute complement is S \ (? ∪ W) = (e, f)
- ∆ symmetric difference
- Set: S = (a, b, c, d)
- Set: T = (a, b, e, f) “(S \ ” ?)∪ (T \ S) = (c, d, e, f
- X Cartesian product
- Set: S = (a, b, c)
- Set: T = (d, e, f)
- S X T = ((a, d), (a, e), (a, f), (b, d), (b, e), (b, f), (c, d), (c, e), (c, f))
- T X S = ((d, a), (e, a), (f, a), (d, b), (b, e), (b, f), (c, d), (c, e), (c, f))
- Note: The Cartesian product operator produces a set of all possible ordered pairs between two sets.
- P(S) Power set
- Set: S = (a, b)
- P(S) = ((a, b), (a), (b), ∅)
- Note: The Power set of S produces the set of all subsets sets including the empty subset.
- between two sets.
- Cardinality
- Set: S = (a, b)
- Cardinality of set S is 2
- Note: The carnality of a set is the number of members in the set.
- Insert
- Set S = (a, b, c)
- Insert(S, d) returns S = (a, b, c, d)
- The Insert operator (i.e. Insert(Set, element)) adds a member to a set.
- Delete
- Set S = (a, b, c)
- Delete(S, a) returns S = (b, c)
- The Delete operator (i.e. Delete(Set, element)) removes a member from a set.
- Lookup
- Set S = (a, b, c)
- Lookup(S, c) returns TRUE
- Lookup(S, d) returns FALSE
- The Lookup operator (i.e. Lookup(Set, element)) determines if an set contains a specific member.
Common Sets:
- ℕ
- The set of all natural numbers (0, 1, 2, 3, 4, 5….)
- ℤ
- The set of all integers (….-3, -2, -1, 0, 1, 2, 3…)
- ℚ
- The set of all rational numbers
- ℝ
- The set of all real numbers