Set

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Basic concept in mathematics and, more specifically, in set theory: a set is a collection of elements that have a particular characteristic in common. The elements are contained or included in the set (i.e. in a relation of ‘inclusion’ to the set) (notation: x ε S, read as ‘x is an element (or member) of S’). Sets can be defined extensionally by naming the number of their elements (enumeration, extension), the order of the elements being insignificant, or intensionally by indicating the common characteristics of the elements (description) ( intension, predicate). In contrast to the everyday language use of the term ‘set,’ mathematical sets have the following characteristics. (a) Concrete objects as well as abstract concepts and mental constructs like numbers, names and phonemes may be elements of sets, which also means that sets, in turn, can be elements of other sets (e.g. the set of all verbs in English is at the same time an element of the set of all word classes in English, if a class is understood as a set of expressions). (b) A set can be empty (empty set, notation: ) (e.g. the set of all clicks in English). (c) A set can consist of a single element (singleton) (e.g. the set of initial symbols in a phrase structure grammar that have only the element S for ‘sentence’ as the initial node). (d) The number of elements of a set can be infinite (e.g. the set of natural numbers or the set of grammatical sentences in English).

The following operations and relations between sets can be distinguished. (e) The identity of sets: two sets A and B are extensionally ‘the same,’ if they contain the same elements. (f) Equivalence: two sets are equivalent, if they can be mapped onto each other bijectively ( function). Equivalence is both a reflexive relation and a symmetric relation and a transitive relation. (g) The union set is that set S to which all elements belong that are included in at least one of the two original sets A and B (notation: =. The union set corresponds in propositional logic to the inclusive, i.e. to the ‘non-exclusive,’ or, the propositional conjunction of which is true if one or both statements are true ( disjunction). See the following Venn diagram for (with hachure):

For example: let A and B be the sets of abstract words and words ending in -ion in English. The union set is, then, the set of abstract words or words ending in -ion in English (billion, carrion, nation, onion, etc.).

(h) The intersection set is the set of those elements that are contained both in set A and in set B (notation: A∩B:= xεB}). For example: if A is the set of transitive and B the set of irregular verbs in English, then the intersection set of A and B is the set of transitive and irregular verbs in English (bind, eat, come). (i) Difference: the difference is that subset of A that contains exactly the same elements in A that are not also elements of B (notation: A\B:= . The union set of the difference A\B and B\A corresponds in propositional logic to the ‘exclusive’ or, the propositional conjunction of which is true only if one of the two statements linked by or is true (but not if both are true) ( disjunction, exclusive disjunction). See the following Venn diagram for A\B (with hachure):

For example: let A be the set of the transitive verbs in English and B the set of irregular verbs in English. The difference A\B is, then, the set of regular transitive verbs in English (e.g. work). (j) Subset: a set A is a subset of a set B if all elements of A are also elements of B (notation: .

For example: the set of transitive verbs in English is a (true) subset of the verbs of the English language, that is to say that in the set of English verbs there is at least one verb that is not an element of the subset of English transitive verbs. In propositional logic the subset corresponds to implication; in semantics ‘subset’ is germane to the relation of hyponymy. (k) Complement: the complement of A with respect to a certain universe of discourse U is the set of all elements that are not elements of A . It is the case that = U\A, that is, the complement of A with respect to U is the special case of the difference of .

For example: let U be the set of all English words. If A is the set of all English verbs, then the complement of set A is the set of all English words except the verbs. (1) Power set: the power set of a set A is the set of all subsets of A (notation: . In this case, the number of elements of the power set corresponds to the number 2 raised to the power of the number of elements in the original set: if A contains the three elements {a, b, c}, then the power set has P(A) 2³=8 elements: , {a}, {b}, {c}, {a, b), {a, c}, {b, c}, {a, b, c}. (m) Disjunction: two sets A and B are disjunct if their intersection (see (h)) yields the empty set , that is, if they do not have any elements in common. Put formally: .

For example: let A be the set of transitive verbs and B the set of intransitive verbs in English; the intersection set is, then, , since no verb can be both intransitive and transitive at the same time. (n) Cartesian product (named for the French philosopher R. Descartes (1596–1650)): the Cartesian product of two sets A and B is the set of all ordered pairs ‹x, y›, wherein x is included in A and y in B, put formally as AXB= and read as ‘A cross B.’

For example: languages with intact inflectional systems use morphological markers for case and number. Let A be the set of grammatical cases in German {nominative, genitive, dative, accusative} and B the set of number {singular, plural}. The Cartesian product of AXB contains all possible combinations {nominative singular, dative plural, etc.}.

set theory

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