Sommerjobb ved FFI INF3580 – Semantic Technologies – Spring 2012 Lecture 5: Mathematical Foundations
Sommerjobber ved Forsvarets Forskningsinstitutt P˚ a Kjeller ved Lillestrøm
Martin Giese
Omtrent 50 studenter ved FFI i 8-10 uker Mange forskjellige fagfelt, bl.a. semantiske teknologier
14th February 2012
Department of Informatics
angi ved søking!
Søknadsfrist: 1. mars Mer info: http://www.ffi.no/no/Om-ffi/Karriere/Sider/Sommerjobb.aspx
University of Oslo
INF3580 :: Spring 2012
Lecture 5 :: 14th February
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Basic Set Algebra
Today’s Plan
Outline
1
Basic Set Algebra
1
Basic Set Algebra
2
Pairs and Relations
2
Pairs and Relations
3
Propositional Logic
3
Propositional Logic
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Basic Set Algebra
Basic Set Algebra
Motivation
Sets: Cantor’s Definition
From the inventor of Set Theory, Georg Cantor (1845–1918): The great thing about Semantic Technologies is. . .
Unter einer “Menge” verstehen wir jede Zusammenfassung M von bestimmten wohlunterschiedenen Objekten m unserer Anschauung oder unseres Denkens (welche die “Elemente” von M genannt werden) zu einem Ganzen.
. . . Semantics! “The study of meaning” RDF has a precisely defined semantics (=meaning)
Translated:
Mathematics is best at precise definitions
A “set” is any collection M of definite, distinguishable objects m of our intuition or intellect (called the “elements” of M) to be conceived as a whole.
RDF has a mathematically defined semantics
There are some problems with this, but it’s good enough for us!
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INF3580 :: Spring 2012
Basic Set Algebra
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Basic Set Algebra
Sets
Elements, Set Equality Notation for finite sets: {‘a’, 1, 4}
A set is a mathematical object like a number, a function, etc. Knowing a set is
Contains ‘a’, 1, and 4, and nothing else. There is no order between elements
knowing what is in it knowing what is not
{1, 4} = {4, 1} Nothing can be in a set several times
There is no order between elements
{1, 4, 4} = {1, 4}
Nothing can be in a set several times Two sets A and B are equal if they contain the same elements
The notation {· · · } allows to write things several times! ⇒ different ways of writing the same thing! We use ∈ to say that something is element of a set:
everything that is in A is also in B everything that is in B is also in A
1 ∈ {‘a’, 1, 4} ‘b’ 6∈ {‘a’, 1, 4} INF3580 :: Spring 2012
{· · · }
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∈ 8 / 41
Basic Set Algebra
Basic Set Algebra
Set Examples {3, 7, 12}:
Know Your Elements! Sets with different elements are different:
a set of numbers
3 ∈ {3, 7, 12}, 0 6∈ {3, 7, 12}
{0}:
{1, 2} = 6 {2, 3}
a set with only one element 0 ∈ {0}, 1 6∈ {0}
{‘a’, ‘b’, . . . , ‘z’}:
What about
a set of letters
{a, b} and {b, c}?
‘y’ ∈ {‘a’, ‘b’, . . . , ‘z’}, ‘æ’ 6∈ {‘a’, ‘b’, . . . , ‘z’},
If a, b, c are variables, maybe
The set P3580 of people in the lecture room right now Martin ∈ P3580 , Albert Einstein 6∈ P3580 .
N = {1, 2, 3, . . .}:
a = 1,
b = 2,
c=1
the set of all natural numbers Then
3580 ∈ N, π 6∈ N.
P = {2, 3, 5, 7, 11, 13, 17, . . .}:
{a, b} = {1, 2} = {2, 1} = {b, c}
the set of all prime numbers
257 ∈ P, 91 6∈ P.
{1, 2, 3} has 3 elements, what about {a, b, c}?
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INF3580 :: Spring 2012
Basic Set Algebra
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Basic Set Algebra
Sets as Properties
The Empty Set
Sets are used a lot in mathematical notation Often, just as a short way of writing things More specifically, that something has a property E.g. “n is a prime number.” In mathematics: n ∈ P E.g. “Martin is a human being”. In mathematics, m ∈ H, where
Sometimes, you need a set that has no elements. This is called the empty set Notation: ∅ or {}
∅
x 6∈ ∅, whatever x is!
H is the set of all human beings m is Martin
One could define Prime(n), Human(m), etc. but that is not usual Instead of writing “x has property XYZ ” or “XYZ (x)”, let P be the set of all objects with property XYZ write x ∈ P. INF3580 :: Spring 2012
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Basic Set Algebra
Basic Set Algebra
Subsets
Set Union The union of A and B contains
Let A and B be sets
A
B
if every element of A is also in B
all elements of A all elements of B also those in both A and B and nothing more.
then A is called a subset of B This is written A⊆B {1} ⊆ {1, ‘a’, 4} {1, 3} 6⊆ {1, 2} P⊆N ∅ ⊆ A for any set A
∪
(A cup which you pour everything into) Examples {1, 2} ∪ {2, 3} = {1, 2, 3} {1, 3, 5, 7, 9, . . .} ∪ {2, 4, 6, 8, 10, . . .} = N ∅ ∪ {1, 2} = {1, 2}
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Basic Set Algebra
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Basic Set Algebra
Set Intersection
Set Difference The set difference of A and B contains
The intersection of A and B contains those elements of A that are also in B and nothing more.
A
those elements of A that are not in B and nothing more.
B
A∩B Examples
B A\B
A\B
∩
\
Examples {1, 2} \ {2, 3} = {1} N \ P = {1, 4, 6, 8, 9, 10, 12, . . .} ∅ \ {1, 2} = ∅ {1, 2} \ ∅ = {1, 2}
{1, 2} ∩ {2, 3} = {2} P ∩ {2, 4, 6, 8, 10, . . .} = {2} ∅ ∩ {1, 2} = ∅
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A
It is written
A∩B
It is written
INF3580 :: Spring 2012
A∪B A∪B
A = B if and only if A ⊆ B and B ⊆ A INF3580 :: Spring 2012
B
It is written
⊆
Examples
A
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Basic Set Algebra
Pairs and Relations
Set Comprehensions
Outline
Sometimes enumerating all elements is not good enough E.g. there are infinitely many, and “. . .” is too vague
1
Basic Set Algebra
2
Pairs and Relations
3
Propositional Logic
Special notation: {x ∈ A | x has some property} The set of those elements of A which have the property. Examples:
{· · · | · · · }
{n ∈ N | n = 2k for some k ∈ N}: the even numbers {n ∈ N | n < 5} = {1, 2, 3, 4} {x ∈ A | x 6∈ B} = A \ B
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Pairs and Relations
Pairs and Relations
Motivation
Pairs A pair is an ordered collection of two objects Written hx, y i
RDF is all about
h· · · i
Equal if components are equal:
Resources (objects) Their properties (rdf:type) Their relations amongst each other
ha, bi = hx, y i
Sets are good to group objects with some properties!
if and only if a = x
and b = y
Order matters: h1, ‘a’i = 6 h‘a’, 1i
How do we talk about relations between objects?
An object can be twice in a pair: h1, 1i hx, y i is a pair, no matter if x = y or not. INF3580 :: Spring 2012
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Pairs and Relations
Pairs and Relations
The Cross Product
Relations A relation R between two sets A and B is. . . . . . a set of pairs ha, bi ∈ A × B R ⊆A×B
Let A and B be sets. Construct the set of all pairs ha, bi with a ∈ A and b ∈ B. This is called the cross product of A and B, written
We often write aRb to say that ha, bi ∈ R Example:
×
A×B
Let L = {‘a’, ‘b’, . . . , ‘z’} Let . relate each number between 1 and 26 to the corresponding letter in the alphabet:
Example: A = {1, 2, 3}, B = {‘a’, ‘b’}. A × B = { h1, ‘a’i , h2, ‘a’i , h1, ‘b’i , h2, ‘b’i ,
h3, ‘a’i , h3, ‘b’i }
1 . ’a’
Lecture 5 :: 14th February
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26 . ’z’
Then . ⊆ N × L:
Why bother? Instead of “ha, bi is a pair of a natural number and a person in this room”. . . . . . ha, bi ∈ N × P3580 But most of all, there are subsets of cross products. . . INF3580 :: Spring 2012
2 . ’b’
. = {h1, ‘a’i , h2, ‘b’i , . . . , h26, ‘z’i} And we can write: h1, ‘a’i ∈ . 21 / 41
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h2, ‘b’i ∈ .
...
h26, ‘z’i ∈ .
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Pairs and Relations
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Pairs and Relations
More Relations
Family Relations Consider the set S = {Homer, Marge, Bart, Lisa, Maggie}. Define a relation P on S such that
A relation R on some set A is a relation between A and A: R ⊆ A × A = A2
xP y
Example:
