What is Soft Computing ? Difference between Soft Computing and Hard Computing. Classical Sets ,operations on classical sets ,Properties of classical sets

1. Chapter 1

2.  Soft computing is an emerging approach to
computing which parallel the remarkable
ability of the human mind to reason and
learn in a environment of uncertainty and
imprecision.

3.  Soft computing differs from conventional
(hard) computing in that, unlike hard
computing
 It is tolerant of imprecision
 Uncertainty
 partial truth
 and approximation
 In effect, the role model for soft computing
is the human mind.

4.  Fuzzy Systems
 Neural Networks
 Evolutionary Computation/Genetic
Algorithm(GA)
 Machine Learning
 Probabilistic Reasoning

5.  The main goal of soft computing is to
develop intelligent machines to provide
solutions to real world problems, which are
not modeled, or too difficult to model
mathematically.
 It’s aim is to exploit the tolerance for
Approximation, Uncertainty, Imprecision, and
Partial Truth in order to achieve close
resemblance with human like decision
making.

6. Hard Computing Soft Computing
 conventional
computing
 Binary logic, crisp
systems, numerical
analysis and crisp
software
 Tolerant to
imprecision,
uncertainty, partial
truth, and
approximation
 fuzzy logic, neural
nets and probabilistic
reasoning.

7. Hard Computing Soft Computing
 requires programs to
be written
 two-valued logic
 Deterministic
 requires exact input
data
 strictly sequential
 precise answers
 can evolve its own
programs
 multivalue or fuzzy
logic
 Stochastic
 Can deal with
ambiguous and noisy
data
 parallel computations
 approximate answers

9. A classical set is defined by crisp boundaries
A fuzzy set is prescribed by vague or ambiguous properties; hence its
boundaries are ambiguously specified
X (Universe of discourse)

10. Classical Set
•A set is defined as a collection of objects,
which share certain characteristics
•A classical set is a collection of distinct objects
-- set of negative integers, set of persons with
height<6 ft, days of the week etc
• Each individual entity in a set is called a
member or an element of the set.
• The Classical set is defined in such a way that
the Universe of Discourse is split into 2 groups:
members and Nonmembers

11. Classical Set
Tuesday
Wednesday
Saturday
Butter
Days of the week

12. Classical Set : Law of the Excluded Middle
•X Must either be in set A or in set not-A,
ie., Of any subject, one thing must be either
asserted or denied
Aristotle

13. Defining a Set
There are several ways of defining a set
•A = {2,4,6,8,10}
•A= {x│x is a prime number <20 }
• A= {xi+1 = (xi +1 )/5, i=1 to 10, where x1=1 }
•A= {x│x is an element belonging to P AND Q }
•µ A(x) = 1 if x ∈ A
= 0 if x ∈ A
Here µ A(x) is membership function for set A

14. •Φ is a null or Empty Set i.e., with no elements
•Set consisting of all possible subsets of a given set
A is called a Power Set
P(A)= {x│x ⊆ A }
•For crisp set A and B containing some elements in
universe X, the notations used are
x ∈ A ⇒ x does belong to A
x ∉ A ⇒ x does not belong to A
x ∈ X ⇒ x does belong to universe X

15. •For classical sets A and B on X we also have
A ⊂ B ⇒ A is completely contained in B
(i.e., if x ∈ A then x ∈ B )
A ⊆ B ⇒ A is contained in or equivalent to B
A=B ⇒ A ⊂ B and B ⊂ A

16. Operations on Classical Sets
Union :
A ∪ B = {x│x ∈ A or x ∈ B }
Intersection :
A ∩ B = {x│x ∈ A and x ∈ B }
Complement :
Ā = {x│x ∉ A , x ∈ X }
Difference:
A-B = A │ B = {x│x ∉ A and x ∉ B }
= A- (A ∩ B ) i.e., All elements in
universe that belong to A but do not belong to B

18. Properties of Classical Sets
Commutivity :
A ∪ B = B ∪ A ; A ∩ B = B ∩ A
Associatively :
A ∪ ( B ∪ C) = ( A ∪ B ) ∪C
A ∩ ( B ∩ C) = ( A ∩ B) ∩C
Distributivity:
A ∪ ( B ∩ C) = ( A ∪ B ) ∩ ( A ∪ C )
A ∩ ( B ∪ C) = ( A ∩ B) ∪ ( A ∩ C )

19. Properties of Classical Sets
Idem potency :
A ∪ A = A ; A ∩ A = A
Transitivity :
If A ⊆ B ⊆ C, then A ⊆ C
Identity:
A ∪ Φ = A ; A ∩ Φ = Φ
A ∪ X = X ; A ∩ X = X

20. Properties of Classical Sets
Law of Excluded middle :
A ∪ Ā = X;
DeMorgan’s Law
Law of contradiction :
A ∩ Ā = Φ;