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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)• …