GEOG 5113 Special Topics in GIScience 5113 Special Topics in GIScience “Fuzzy Set Theory in...
Transcript of 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
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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)
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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
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A"B = B"A and A#B = B#A
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(A"B)"C = A" (B"C) and (A#B)#C = A# (B#C)
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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
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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
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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
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"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
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"A (x) = 1# "
A(x)
Characteristic functions: Union
• C.F. of Union of A and B
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"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
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"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)• …