geog 5113 special topics in giscience 5113 special topics in giscience “fuzzy set theory in...

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GEOG 5113Special Topics in GIScience

“Fuzzy Set Theory in GIScience”

-Classical Set Theory-

Classic, Crisp and Sharp

• As for classic logic we assume we canmake (crisp, exact) distinctionsbetween and among groups

• Groups or sets with sharp boundaries• An individual is definitely in or out

Set

• Most basic concept in logic and mathematics• Any collection of items or individuals• Collections: Anything! (Cars, buildings,

students)• Things that can be distinguished from one

another as individuals and that share someproperty

• ‘a’ is a Member or element of the set ‘A’: a ∈ A• Only two possible relationships between a and

A: ∈ or ∉

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Standard symbols

• Universal proposition∀a ∈ A -- “for any element a in set A”

• Existential proposition∃a ∈ A -- “there exists at least one element ain set A”

• “Such that” ∃a ∈ A | a>3 “… such that a is greater than 3.”

Representation of Sets• Representation of a set as list A = {a,b,c}• Number of members of a finite set is its size and is

called CARDINALITY: |A| = 3 (if |A| = 0: Singleton)• Representation of a set using the rule method:

C = {x|P(x)}• “the set C is composed of elements x, such that

(every) x has the property P”• Proposition P(x) is either true or false for any given

individual xE = {x | x is a legal United States coin}

Set families

• A set whose members are setsthemselves is referred to a “family ofsets”

• {Ai | i ∈ I}• i: index; I: index set• Families of sets: A, B, C

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Universal and Empty Set

• Universal set X consists of all theindividuals that are of interest in thatapplicationE.g., classifying all students on campusX consists of all students on campus

• The empty set ∅ is a set that containsnothing at all

Set inclusion• A is called a subset of B if every member of set A is

also a member of set B:A ⊆ B(every set is a subset of itself)

• Venn diagrams• If A ⊆ B and B ⊆ A then A=B (equal sets)• If A ⊆ B and A ≠B then B contains at least one

element that is not a member of A. A is a propersubset of B:A ⊂ B

• ∅ ⊆ A ⊆ X

Power Set

• Set which contains all possible subsets of agiven universal set X: P(X)

• P(X) is an abbreviation for {A | A ⊆ X} or {A | A ∈P(X)}

• If |X| = n, then the number of possible subsets |P(X)|= 2n (two possibilities for each element of X)X = {a,b,c}Try to find out: Number of possible subsets(combinations of members, basically)

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P(X) = {∅,{a},{b},{c},{a,b},{a,c},{b,c},X}

Set Operations

• Complement

• Union

• Intersection

• Difference

Complement & Union• Complement

Set of all elements in X that are not in A

!

A = {x | x " X and x # A}

!

X =" ; A = A (involution)

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Union• Union

All elements that belong to either A or B, or to both(union of a set with its complement is X); disjunctionLaw of excluded middle: All elements of the universalset X must belong to either a set A or its complement

!

A"B = {x | x # A or x # B}

!

A"A = X

Intersection & Difference

• IntersectionAll elements that belong to A and B simultaneously(conjunction). Elements have properties of both sets.Law of contradiction:(A set A and its complement do not overlap!; thesame for “disjoint” sets)

!

A"B = {x | x # A and x # B}

!

A"A =#

Difference

• DifferenceAll elements that belong to A but not to B

!

A " B = {x | x # A and x $ B}

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Properties of Combined Sets

• Involution• Law of contradiction• Law of excluded middle

• Commutativity, Associativity, Idempotence• Distributivity• DeMorgan’s Law

Do not hold forFuzzy Sets

!

A = A

!

A"A =#

!

A"A = X

Commutativity, Associativity,Idempotence

• Order does not matter for union and intersection(Commutative)

• If more than 2 sets are combined with only union oronly intersection operators, the placement ofparentheses - grouping any two sets together - hasno effect, order does not matter! (associative)

• Union and intersection of a set with itself yields theoriginal set (idempotency) to collapse redundantstrings

!

A"B = B"A and A#B = B#A

!

(A"B)"C = A" (B"C) and (A#B)#C = A# (B#C)

!

A"A = A and A#A = A

Distributivity

• Law of Distribution• Distribute a set on one side of a union

operator over the intersection of twoother sets and vice versa.

• Original main operator and originalsubsidiary operator both become theiropposites

!

A" (B#C) and (A"B)# (B"C)

A# (B"C) and (A#B)" (B#C)

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De Morgan’s Law

• Transformation of intersection intounions, and vice versa, by dealing withtheir complements

• Complement of intersection (union) oftwo sets is equivalent to the union(intersection) of their individualcomplements

• Try to combine with involution

!

A"B = A #B

A#B = A "B

Characteristic Functions ofCrisp Sets

• Function is an assignment of elements of one set A toelements of another set B

• Elements of B are images or values of elements of A• A = {a,b,c} is a set with 3 members; B = {F,T} is a second

set (B = {0,1})• When stipulating truth values of each of the three

propositions a,b,c we assign to each member of A anelement of B (truth values)

• Every element in A must be assigned an element in B• Each element in A can be assigned only one element

in B

Characteristic Functions

• Function f from set A to set B is: A→B• Many-to-one function• One-to-one function• Let A be a subset of X. Then its characteristic

function is defined for each x ∈ X by:• Each element is IN or OUT

!

"A =1 if x # X

0 if x $ X

% & '

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Example

• CHARACTERISTIC FUNCTION OFTHE SET OF REAL NUMBERS FROM5 TO 10

!

"A =1 if 5 # x # 10

0 otherwise

$ % &

Subset & Set operationsrepresented functionally

• A is a subset of B if …:

• Characteristic function of thecomplement of a set A!

A " B if and only if #A (x) $ #B (x) for each x % X

!

"A (x) = 1# "

A(x)

Characteristic functions: Union

• C.F. of Union of A and B

!

"A#B (x) = max("

A(x),"

B(x) )

Figs. 3.10, 3.11

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Characteristic functions: Intersection

• C.F. of Intersection of A and B

!

"A#B (x) = min("

A(x),"

B(x) )

Some further concepts

• Set of Real Numbers: R• X-axis (real line/axis): One dimensional

Euclidean space• Intervals (closed, oben, half open)• …