Mathematical Methods in Linguistics. Basic Concepts of Set Theory.
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Transcript of Mathematical Methods in Linguistics. Basic Concepts of Set Theory.
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Mathematical Methods in Linguistics
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Basic Concepts of Set Theory
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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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FST - Torbjörn Lager, UU 4
Specification of Sets
List notation (enumeration)DiagramPredicate notationRecursive rules
For an example, see page 9 in MML
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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"
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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
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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}
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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
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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
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FST - Torbjörn Lager, UU 10
Set Theoretic Equalities
See page 18 in MML
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Relations and Functions
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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
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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
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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
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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
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Properties of Relations
page 39-53 in MMLthis part is optional
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Lecture 2:Logic and Formal Systems
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FST - Torbjörn Lager, UU 18
Basic Concepts of Logic and Formal Systems
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FST - Torbjörn Lager, UU 19
Statement Logic
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FST - Torbjörn Lager, UU 20
Predicate Logic
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Lecture 3:Knowledge and Meaning Representation
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Lecture 4:English as a Formal Language
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FST - Torbjörn Lager, UU 23
Compositionality
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FST - Torbjörn Lager, UU 24
Lambda Abstraction
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Lecture 5:Finite Automata, Regular Languages and Type 3 Grammars
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Lecture 6:Pushdown Automata, Context Free Grammars and Languages
Lecture 6:Feature Structures and Equations
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Lecture 7:Feature Structures and Unification-Based Grammars