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Mathematical Methods in Linguistics
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Mathematical Methods in Linguistics. Basic Concepts of Set Theory.

Dec 19, 2015

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Page 1: Mathematical Methods in Linguistics. Basic Concepts of Set Theory.

Mathematical Methods in Linguistics

Page 2: Mathematical Methods in Linguistics. Basic Concepts of Set Theory.

Basic Concepts of Set Theory

Page 3: 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)

Page 4: Mathematical Methods in Linguistics. Basic Concepts of Set Theory.

FST - Torbjörn Lager, UU 4

Specification of Sets

List notation (enumeration)DiagramPredicate notationRecursive rules

For an example, see page 9 in MML

Page 5: Mathematical Methods in Linguistics. Basic Concepts of Set Theory.

FST - Torbjörn Lager, UU 5

Identity and Cardinality

Identity

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

Cardinality

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

Page 6: Mathematical Methods in Linguistics. Basic Concepts of Set Theory.

FST - Torbjörn Lager, UU 6

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

Page 7: Mathematical Methods in Linguistics. Basic Concepts of Set Theory.

FST - Torbjörn Lager, UU 7

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}

Page 8: Mathematical Methods in Linguistics. Basic Concepts of Set Theory.

FST - Torbjörn Lager, UU 8

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

Page 9: Mathematical Methods in Linguistics. Basic Concepts of Set Theory.

FST - Torbjörn Lager, UU 9

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

Page 10: Mathematical Methods in Linguistics. Basic Concepts of Set Theory.

FST - Torbjörn Lager, UU 10

Set Theoretic Equalities

See page 18 in MML

Page 11: Mathematical Methods in Linguistics. Basic Concepts of Set Theory.

Relations and Functions

Page 12: Mathematical Methods in Linguistics. Basic Concepts of Set Theory.

FST - Torbjörn Lager, UU 12

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

Page 13: Mathematical Methods in Linguistics. Basic Concepts of Set Theory.

FST - Torbjörn Lager, UU 13

Functions: Domain and Range

rop

trometa

jul

ful

mat

tafå

feg

be

klo

se

Domain

4

3

2

Range

5

61

Page 14: Mathematical Methods in Linguistics. Basic Concepts of Set Theory.

FST - Torbjörn Lager, UU 14

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

Page 15: Mathematical Methods in Linguistics. Basic Concepts of Set Theory.

FST - Torbjörn Lager, UU 15

Functions (cont'd)

rop

trometa

jul

ful

mat

tafå

feg

be

klo

se

Domain

4

3

2

Range

5

6

Page 16: Mathematical Methods in Linguistics. Basic Concepts of Set Theory.

Properties of Relations

page 39-53 in MMLthis part is optional

Page 17: Mathematical Methods in Linguistics. Basic Concepts of Set Theory.

Lecture 2:Logic and Formal Systems

Page 18: Mathematical Methods in Linguistics. Basic Concepts of Set Theory.

FST - Torbjörn Lager, UU 18

Basic Concepts of Logic and Formal Systems

Page 19: Mathematical Methods in Linguistics. Basic Concepts of Set Theory.

FST - Torbjörn Lager, UU 19

Statement Logic

Page 20: Mathematical Methods in Linguistics. Basic Concepts of Set Theory.

FST - Torbjörn Lager, UU 20

Predicate Logic

Page 21: Mathematical Methods in Linguistics. Basic Concepts of Set Theory.

Lecture 3:Knowledge and Meaning Representation

Page 22: Mathematical Methods in Linguistics. Basic Concepts of Set Theory.

Lecture 4:English as a Formal Language

Page 23: Mathematical Methods in Linguistics. Basic Concepts of Set Theory.

FST - Torbjörn Lager, UU 23

Compositionality

Page 24: Mathematical Methods in Linguistics. Basic Concepts of Set Theory.

FST - Torbjörn Lager, UU 24

Lambda Abstraction

Page 25: Mathematical Methods in Linguistics. Basic Concepts of Set Theory.

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

Page 26: Mathematical Methods in Linguistics. Basic Concepts of Set Theory.

Lecture 6:Pushdown Automata, Context Free Grammars and Languages

Lecture 6:Feature Structures and Equations

Page 27: Mathematical Methods in Linguistics. Basic Concepts of Set Theory.

Lecture 7:Feature Structures and Unification-Based Grammars