Fuzzy Set Operations

1
Fuzzy Set Operations

• If the membership value of all elements is
  constrained to either 1 or 0 then we have a
  crisp' or boolean set.
• e.g
• (Tom, 0), (Mary, 1), (Jack, 0), (Jill, 0) means
  the same as Mary
• Any operators for fuzzy set union (OR) and
  intersection (AND) must exhibit the following
  properties
• generalises to crisp (or boolean sets)
• a decrease in membership of an element in each
  of the fuzzy sets cannot lead to an increase in
  the membership of that element in the
  intersection or the union
• the order doesn't matter (i.e. it is
  commutative)
• the operators can be applied in a pairwise
  fashion for any number of sets (i.e. it is
  associative)
• The min operator for intersection (AND) and the
  max operator for union (OR) exhibit these
  properties

2
Fuzzy Set Operations

• for 0 lt a,b,c,d lt 1
• min(0,0) 0, max(1,1) 1
• min(a,1) a, max(a,0) a cannot generate out
  of range values
• min(a,b) lt min(c,d) when altc and b lt d
• max(a,b) lt max(c,d) when altc and b lt d
• min(a,b) min(b,a)
• max(a,b) max(b,a) commutative law
• min(min(a,b),c) min(a, min(b,c))
• max(max(a,b),c) max(a, max(b,c)) associative
  law

3
Fuzzy Set Operations

• membership of each element in the intersection (
  AND) of two fuzzy sets minimum membership
• short people (Tom,0), (Mary,1), (Jack,0.2),
  (Jill,0.7)
• average people (Tom,0.8), (Mary,0),
  (Jack,0.8), (Jill,0.3)
• so the set of short people and average people
• (Tom,0), (Mary,0), (Jack,0.2),
  (Jill,0.3)
• membership of each element in the union ( or) of
  two fuzzy sets maximum membership
• so the set of short people or average people
• (Tom,0.8), (Mary,1), (Jack,0.8),
  (Jill,0.7)

4
Fuzzy Set Operations

• Note These operators are a generalisation of
  the boolean operators
• short people (Tom,0), (Mary,1), (Jack,0),
  (Jill,1)
• young people (Tom,1), (Mary,1), (Jack,0),
  (Jill,1)
• short people and young people
• (Tom,0), (Mary, 1), (Jack,0), (Jill,1)
• short people or young people
• (Tom,1), (Mary, 1), (Jack,0), (Jill,1)

• In boolean terms
• short people Mary, Jill
• young people Tom, Mary, Jill
• short people n young people Mary, Jill
• short people U young people Mary, Jill, Tom

5
Fuzzy Set Operations

• Set complement
• membership of each element in the complement
  (not) of a fuzzy set 1 - membership
• Using the previous example
• If short people (Tom,0), (Mary,1),
  (Jack,0.2), (Jill,0.7) then
• not short people (Tom,1), (Mary,0),
  (Jack,0.8), (Jill,0.3)
• Again this is a generalisation of the boolean
  case i.e.
• short people (Tom,0), (Mary,1), (Jack,0),
  (Jill,1) or mary, jill
• not short people (Tom,1), (Mary,0), (Jack,1),
  (Jill,0) or tom, jack

6
Fuzzy Set Operations

• Set difference
• If A and B are fuzzy sets then how to evaluate
  membership in A\B?
• A\B A and (not B)
• Take an example
• short people (Tom,0), (Mary,1), (Jack,0.2),
  (Jill,0.7)
• average people (Tom,0.8), (Mary,0),
  (Jack,0.6), (Jill,0.5) then
• short people\average people short people and
  (not average people)
• (Tom,0), (Mary,1), (Jack,0.2), (Jill,0.7) and
  (Tom,0.2), (Mary,1), (Jack,0.4), (Jill,0.5)
• (Tom,0), (Mary,1), (Jack,0.2), (Jill,0.5)
• It can be shown that again this is a
  generalisation of the boolean case

7
Fuzzy Set Operations

• Summary
• A fuzzy set can be defined as a function from
  some reference set to the real number interval
  0,1 i.e. reference set ---gt 0,1
• membership of a reference set element in the
  fuzzy set union (OR) ---- use the max operator
• membership of a reference set element in the
  fuzzy set intersection (AND) ---- use the min
  operator
• membership of a reference set element in the
  complement of a fuzzy set (NOT) ---- use one
  minus the membership value
• membership of a reference set element in the
  difference between two fuzzy sets (A and B) ----
  use minmembership in A, 1 - membership in B

8
Fuzzy sets

• Some curious and interesting results
• Does the basic law, A n (not A) empty set,
  still hold?
• In fuzzy set theory the empty set is one where
  the membership of all reference set elements is 0
• i.e (Tom,0), (Mary,0), (Jack,0), (Jill,0) is
  the empty set for the previous example
• short people (Tom,0), (Mary,1), (Jack,0.2),
  (Jill,0.7)
• not short people (Tom,1), (Mary,0),
  (Jack,0.8), (Jill,0.3)
• short people and (not short people) (Tom,0),
  (Mary,0), (Jack,0.2), (Jill,0.3)
• which is not the empty set so the law does not
  hold
• In other words an elements can show some
  membership in a set and in its complement
  simultaneously.

9
Fuzzy sets

• Continuous and discrete fuzzy sets
• In all of the previous examples the reference set
  for the fuzzy set is discrete
• The set elements and the membership can be
  enumerated (ie. listed')
• In many cases the reference set may be continuous
  e.g. the set of real numbers (R).
• So the membership function will be continuous.
• short_people R ---gt 0,1 where

1
M'ship
0
Height (cm)
150
170
10
Fuzzy sets

• Continuous and discrete fuzzy sets
• average_people R ---gt0,1
• tall_people R ---gt0,1
• Although continuous these sets can be bounded
  (i.e. finite)

average_people
tall_people
1
M'ship
0
Height (cm)
150
170
190
11
Fuzzy sets

• Continuous and discrete fuzzy sets
• The fuzzy set operators can be applied to these
  sets
• average_people AND short_people

average_people
tall_people
short_people
1
M'ship
0
Height (cm)
150
170
190
12
Fuzzy sets

• Continuous and discrete fuzzy sets
• The fuzzy set operators can be applied to these
  sets
• average_people AND short_people

tall_people
1
average_people AND short_people
M'ship
0
Height (cm)
150
170
190
13
Fuzzy sets

• Continuous and discrete fuzzy sets
• The fuzzy set operators can be applied to these
  sets
• average_people OR short_people

average_people OR short_people
tall_people
1
M'ship
0
Height (cm)
150
170
190
14
Fuzzy sets

• Continuous and discrete fuzzy sets
• The fuzzy set operators can be applied to these
  sets
• NOT tall_people

NOT tall_people
tall_people
1
M'ship
0
Height (cm)
150
170
190
15
Fuzzy sets

• Comments
• In each case there is at least one reference set
  element that exhibits total membership in the
  set.
• such sets are referred as being normal
• The total membership of any given reference set
  element across the three sets is always 1. This
  is a desired but not a necessary property when
  using fuzzy sets to model a linguistic variable