## Sets and Individuals

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| Sets and Individuals in CYC®
| CYC® Predicates

The CYC® KB is a set of assertions about the world. Those assertions may be stated as expressions in CycL, the CYC® representation language. The terms of CycL expressions can be variables, certain kinds of objects native to the computational substrate (such as strings or integers), or CYC® constants. CYC® contains objects in the KB which are created to denote particular concepts. They have unique names and are written with the prefix "#$".

CYC® constants can either denote sets, like "the set of all people", or individuals, like "the Eiffel Tower". Every term in CYC® is an element of #$Thing, the universal set. #$Thing is partitioned into #$Individual and #$SetOrCollection.

#$Individual denotes the set of all things which are not sets. Individuals in the CYC® KB include constants such as #$CityOfSanFrancisco and #$JimmyCarter.

#$SetOrCollection is partitioned into #$Set-Mathematical and #$Collection. The distinction between mathematical sets and collections in CYC® is an important one. Both sets and collections can have elements. Thus, they can both enter into set-theoretic relations like "subset" and "superset".

The difference between #$Collection and #$Set-Mathematical is that sets, in the mathematical sense, are defined extensionally -- that is, by their members. Two sets that have the same members are equivalent. CYC® collections, on the other hand, are defined intensionally, by their criteria for membership. So in CYC®, #$USPresidentsNamedRoosevelt, #$USPresidentsWhoWereEachOthersFifthCousins, and #$TheFirstTwoUSPresidentsWhoseLastNamesBeginWithR would all be different #$Collections, even though they are all comprised of exactly the same elements as the mathematical set {Theodore Roosevelt, Franklin Delano Roosevelt}.

In CYC®, collections are much more important, and much more commonly used, than sets. The criteria for membership in a #$Collection need not be stated explicitly in the KB. Collections in the CYC® KB include #$Dog (the collection of all dogs), #$RealNumber (the collection of all real numbers), and #$Buying (the collection of all buying events).

Membership in a collection is typically expressed as "instance of" or "element of" or "is a", as in "Fred is an instance of the collection #$Person," or "Fred is an element of #$Person," or "Fred is a #$Person." If the terms "subset" and "superset" are used with reference to collections, they typically are intended to mean "more specific collection" and "more general collection", respectively.

Last Update: 03/28/2002