Mathematical Methods in Linguistics

Basic Concepts of Set Theory

FST - Torbjörn Lager, UU 3

What Is a Set?

An abstract collection of distinct object (its members)

Can have (almost) anything as a member, including other sets

May be small (even empty) or large (even infinite)

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Specification of Sets

List notation (enumeration)DiagramPredicate notationRecursive rules

For an example, see page 9 in MML

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Identity and Cardinality

Identity

{Torbjörn Lager} = {x | x is the teacher in C389}

Cardinality

|A| means "the number of elements in the set A"

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The Member and Subset Relations

a A means "a is a member of the set A"A B means "every element of A is also an

element of B"A 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"

a B means a B does not holdA B means A B does not hold

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Powerset

The powerset of a set A is the set of all subsets of A

E.g the powerset of {a,b} is {{a,b},{a},{b},Ø}

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Union and Intersection

The 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)

The 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

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Difference and Complement

The difference between two sets A and B, written A-B, is all the elements of A which are not also elements of B

The complement of a set A and B, written A', is all the elements which are not in A

A complement of a set is always relative to a universe U. It also holds that A' = U-A

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Set Theoretic Equalities

See page 18 in MML

Relations and Functions

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Ordered Pairs and Cartesian Products

The Cartesian product (sv: "kryssprodukten") of A and B, written A B, is the set of pairs <x,y> such that x is an element in A and y is an element in B

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Functions: Domain and Range

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Domain

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Range

5

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A Function

A set of pairs

Each element is in the domain is paired with just one element in the range

A 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

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Functions (cont'd)

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Domain

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Range

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Properties of Relations

page 39-53 in MMLthis part is optional

Lecture 2:Logic and Formal Systems

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Basic Concepts of Logic and Formal Systems

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Statement Logic

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Predicate Logic

Lecture 3:Knowledge and Meaning Representation

Lecture 4:English as a Formal Language

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Compositionality

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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