GEOG 5113
Special Topics in GIScience
“Fuzzy Set Theory in GIScience”
-Classical Set Theory-
Classic, Crisp and Sharp
• As for classic logic we assume we can
make (crisp, exact) distinctions
between 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 some
property
• ‘a’ is a Member or element of the set ‘A’: a ∈ A
• Only two possible relationships between a and
A: ∈ or ∉
1
Standard symbols
• Universal proposition
∀a ∈ A -- “for any element a in set A”
• Existential proposition
∃a ∈ A -- “there exists at least one element a
in 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 x
E = {x | x is a legal United States coin}
Set families
• A set whose members are sets
themselves is referred to a “family of
sets”
• {Ai | i ∈ I}
• i: index; I: index set
• Families of sets: A, B, C
2
Universal and Empty Set
• Universal set X consists of all the
individuals that are of interest in that
application
E.g., classifying all students on campus
X consists of all students on campus
• The empty set ∅ is a set that contains
nothing 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 proper
subset of B:
A⊂B
• ∅⊆A⊆X
Power Set
• Set which contains all possible subsets of a
given 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)
3
P(X) = {∅,{a},{b},{c},{a,b},{a,c},{b,c},X}
Set Operations
• Complement
• Union
• Intersection
• Difference
Complement & Union
A = {x | x " X and x # A}
• Complement
Set of all elements in X that are not in A
X = " ; A = A (involution)
!
!
4
Union
• Union
All elements that belong to either A or B, or to both
(union of a set with its complement is X); disjunction
Law of excluded middle: All elements of the universal
set X must belong to either a set A or its complement
A " B = {x | x # A or x # B}
A"A = X
!
!
Intersection & Difference
A " B = {x | x # A and x # B}
• Intersection
All 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!; the
same for “disjoint” sets)
A"A =#
!
Difference
• Difference
All elements that belong to A but not to B
A " B = {x | x # A and x $ B}
!
5
Properties of Combined Sets
• Involution
A=A
A"A =#
• Law of contradiction
• Law of excluded middle A " A = X
Do not hold for
Fuzzy Sets
!
• Commutativity, Associativity, Idempotence
!
• Distributivity !
• DeMorgan’s Law
Commutativity, Associativity,
Idempotence
• Order does not matter for union and intersection
(Commutative)
A " B = B " A and A # B = B # A
• If more than 2 sets are combined with only union or
only intersection operators, the placement of
parentheses - grouping any two sets together - has
no effect, order does not matter! (associative)
!
(A " B) "C = A " (B "C) and (A # B) #C = A # (B #C)
• Union and intersection of a set with itself yields the
original set (idempotency) to collapse redundant
strings
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 two
other sets and vice versa.
• Original main operator and original
subsidiary operator both become their
opposites
A " (B #C) and (A " B) # (B "C)
A # (B "C) and (A # B) " (B #C)
!
6
De Morgan’s Law
• Transformation of intersection into
unions, and vice versa, by dealing with
their complements
• Complement of intersection (union) of
two sets is equivalent to the union
(intersection) of their individual
complements
A "B = A #B
• Try to combine with
involution
A #B = A "B
!
Characteristic Functions of
Crisp Sets
• Function is an assignment of elements of one set A to
elements 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 an
element 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:
%1 if x # X
• Each element is IN or OUT
"A = &
' 0 if x $ X
!
7
Example
• CHARACTERISTIC FUNCTION OF
THE SET OF REAL NUMBERS FROM
5 TO 10
$1 if 5 # x # 10
"A = %
& 0 otherwise
!
Subset & Set operations
represented functionally
• A is a subset of B if …:
A " B if and only if # A (x) $ # B (x) for each x % X
• Characteristic function of the
complement of a set A
" 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
8
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)
• …
9