Mathematical Methods in Linguistics

Basic Concepts of Set Theory

FST - Torbjörn Lager, UU 3 What Is a Set? zAn abstract collection of distinct object (its members) zCan have (almost) anything as a member, including other sets zMay be small (even empty) or large (even infinite)

FST - Torbjörn Lager, UU 4 Specification of Sets zList notation (enumeration) zDiagram zPredicate notation zRecursive rules zFor an example, see page 9 in MML

FST - Torbjörn Lager, UU 5 Identity and Cardinality zIdentity y{Torbjörn Lager} = {x | x is the teacher in C389} zCardinality y|A| means "the number of elements in the set A"

FST - Torbjörn Lager, UU 6 The Member and Subset Relations za  A means "a is a member of the set A" zA  B means "every element of A is also an element of B" zA  B means "every element of A is also an element of B and there is at least one element of B which is not in A" za  B means a  B does not hold zA  B means A  B does not hold

FST - Torbjörn Lager, UU 7 Powerset zThe powerset of a set A is the set of all subsets of A zE.g the powerset of {a,b} is {{a,b},{a},{b},Ø}

FST - Torbjörn Lager, UU 8 Union and Intersection zThe union of two sets A and B, written A  B, is the set of all objects that are members of either the set A or the set B (or both) zThe intersection (sv: "snittet") of two sets A and B, written A  B, is the set of all objects that are members of both the set A and the set B

FST - Torbjörn Lager, UU 9 Difference and Complement zThe difference between two sets A and B, written A-B, is all the elements of A which are not also elements of B zThe complement of a set A and B, written A', is all the elements which are not in A zA complement of a set is always relative to a universe U. It also holds that A' = U-A

FST - Torbjörn Lager, UU 10 Set Theoretic Equalities zSee page 18 in MML

Relations and Functions

FST - Torbjörn Lager, UU 12 Ordered Pairs and Cartesian Products zThe Cartesian product (sv: "kryssprodukten") of A and B, written A  B, is the set of pairs such that x is an element in A and y is an element in B

FST - Torbjörn Lager, UU 13 Functions: Domain and Range rop tro meta jul ful mat ta få feg be klo se Domain Range 5 6 1

FST - Torbjörn Lager, UU 14 A Function zA set of pairs zEach element is in the domain is paired with just one element in the range zA subset of a Cartesian product A  B can be called a function just in case every member of A occurs exactly once a the first element in a pair

FST - Torbjörn Lager, UU 15 Functions (cont'd) rop tro meta jul ful mat ta få feg be klo se Domain Range 5 6

Properties of Relations page in MML this part is optional

Lecture 2: Logic and Formal Systems

FST - Torbjörn Lager, UU 18 Basic Concepts of Logic and Formal Systems

FST - Torbjörn Lager, UU 19 Statement Logic

FST - Torbjörn Lager, UU 20 Predicate Logic

Lecture 3: Knowledge and Meaning Representation

Lecture 4: English as a Formal Language

FST - Torbjörn Lager, UU 23 Compositionality

FST - Torbjörn Lager, UU 24 Lambda Abstraction

Lecture 5: Finite Automata, Regular Languages and Type 3 Grammars

Lecture 6: Pushdown Automata, Context Free Grammars and Languages Lecture 6: Feature Structures and Equations

Lecture 7: Feature Structures and Unification-Based Grammars