Sets {universal set}| can contain everything.
Sets {null set} {empty set} can contain nothing.
Sets {countable set} can be in one-to-one correspondence with positive-integer set.
Two sets {disjoint set, non-overlapping} {non-overlapping set} can have no members in common.
Sets {equivalence class} can include all number examples. Equivalence classes are typically infinite sets. For example, the equivalence class of -2 can include all possible pairs of natural-number subtractions {equivalence relation, class}: -2 = 0 - 2 = (0,2) and -2 = 1 - 3 = (1,3), and so on. Complex numbers are equivalence classes of remainders of polynomials x / (x^2 + 1), where x is a real number.
All congruent integers form set {residue class, set}, [x]n or Zn.
Sets {closed set, interval} can contain limit point at boundary.
Sets {derived set} can have other-set limit points.
Sets {open set} can contain no boundary points and so have only interior points.
All closed-set {perfect set} points can be at the limit.
Set systems {monotonic set} can have previous sets contained in next sets.
Sets {ordered set} can have elements that follow trichotomy and transitivity relations. Sets {partially ordered set} can have elements that follow trichotomy and transitivity relations.
Relation between second coordinate and first coordinate can be same as relation between first coordinate and second coordinate {reflexive set}.
Relation between second coordinate and first coordinate can be from higher to lower {transitive set}.
Subsets {complement, set}| {absolute complement} can have all set members that do not belong to another subset. Subsets {relative complement} can have set members that do not belong to another subset and belong to third subset.
Subsets {coset, subset} can contain products of one set element times all set elements.
Sets {nested set} {nest, set} can have one set inside the other.
For a set, a set {power set} can contain all subsets. For n-element sets, power sets have 2^n elements. Subset number is greater than set-member number.
Outline of Knowledge Database Home Page
Description of Outline of Knowledge Database
Date Modified: 2022.0225