Union of set a and its intersection with another set b is the set {absorption law}: a V (a & b) = a. Intersection of set a and its union with another set b is the set: a & (a V b) = a.
Union of set a with itself is the set {idempotent law}: a V a = a. Intersection of set a with itself is the set: a & a = a.
Absolute complement of set union equals intersection of set absolute complements {De Morgan's laws, set} {De Morgan laws, set} {De Morgan's complement}: ~(a V b) = ~a & ~b. Absolute complement of set intersection equals union of set absolute complements: ~(a & b) = ~a V ~b.
Absolute complement of set absolute complement is set {involution, set}: ~(~a) = a. Absolute complement of empty set is universal set. Absolute complement of universal set is empty set.
Sets have types {theory of types, laws of form} {laws of form}. Sets have lower or higher type than other sets. Lower sets cannot be in statements about higher sets, and higher sets cannot be in statements about lower sets.
differences
Laws of form use differences {calculus of indications}.
self-reference
In set type, boundary contains object surrounded, and object is boundary {self-reference, set}. Self-reference allows references to members, classes, classes of classes, and so on. Imaginary numbers express self-reference and self-referential statements, because imaginary numbers can represent time dimensions.
Statements that use set types correctly can use self-reference. Statements that do not use set types correctly must not use self-reference.
space or set
Members, definition, description, selection, name, or statement define space or set {space, laws of form}. Spaces can imply boundaries and have natural boundaries. Absolute time and relative time can be boundaries and show boundary history {creativity, laws of form}.
logical laws
Laws of form are equivalent to logical laws. NOT a = marked a. a OR b = NOT a AND NOT b. a AND b = NOT (NOT a OR NOT b). a THEN b = NOT a OR b.
Sets have open or closed boundaries {boundary, set}. Actions can create a boundary {expansion, laws of form} or cross a boundary {contraction, laws of form}. Complex actions combine expansions and contractions.
marking
Boundary side can be positive {marked region} and other side negative {unmarked region}.
space
Drawing boundary inside space tells nothing about whole space. Encircling whole space with boundary tells nothing about whole space.
boundary making
Interaction between observer and system {boundary making} makes larger system containing both observer and original system. Observer can surround all or some sets or be inside a set in the set hierarchy.
Observer cannot know whole system, only part close to boundary. Observer makes boundaries to describe whole system.
interval
Regions {interval, laws of form} have boundaries. Boundaries can be in marked sets or spaces {open interval, set}. Boundaries can not be in marked sets or spaces {closed interval, set}.
operations at boundaries
Going from inside boundary to outside boundary is opposite operation {inverse, laws of form}. Going from inside boundary to outside boundary, and then going from inside boundary to outside boundary, results in original condition {identity, laws of form}. Drawing same boundary second time is NULL operation and gives no new information.
Two mutually exclusive same-type sets can become one same-type set, by crossing a boundary. Statements {contraction statement} can name two classes {categorizing, laws of form} and a relation {description, laws of form}, such as equality, with a reference point from which to relate members. Instructions {selection rule, laws of form} can be about how to select members. Processes can cross boundaries.
Sets can divide into two mutually exclusive same-type sets, by creating a boundary. Statements {expansion statement} can name {naming, laws of form} classes {definition, expansion} and a distinction, such as equality, with a reference point from which to identify members. Instructions {selection rule, expansion} can be about how to select members. Processes can draw boundaries.
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Date Modified: 2022.0225