Soft computing

1. Soft computing
Crisp relations
By,
S.vijayalakshmi.

2. Crisp relations:
 A subset of cartesian product A1*A2*…*Ar is called an r-r-any
relation over A1,A2,…Ar . Again the most common Case is for
r=2;inthis situation , the relation is a subset of cartesian product
A1*A2.
 The cartesian product of two universes X and Y is determined as
X*Y={(X,Y)|xЄX , yЄY}

3. Cartesian product:
 The Cartesian product of two sets A & B is denoted by A X B and
is the set of all ordered pairs such that the first element in the pair
belongs to set A and second element belong to set B.
A*B={(a , b) /aЄA , bЄB}
 It can be observed that cardinality of A X B is the product of
cardinality of individual sets.
 A = { 1, 2, 3 }
 B = { a, b }
 A X B = { (1, a), (1, b), (2, a), (2, b), (3, a), (3, b) }

4. Example
A1={a , b} , A2={1 , 2}, A3={α }
A1*A2={(a,1),(b,1),(a,2),(b,2)},|A1*A2|=4 and
|A1|=|a2|=2
|A1*A2|=|A1|.|A2|
A1*A2*A3={(a,1,α),(a,2,α ),(b,2,α)}
|A1*A2*A3|=4=|A1|.|A2|.|A3|

5. Other Crisp Relations :-
 Any crisp relation R (x1, x2, x3………….xn) among crisp sets x1,
x2, x3,…………….,xn is a subset of the Cartesian product.
 for n = 2 the relation R(x1, x2) is called binary relation.
 for n = 3 the relation R(x1, x2, x3) is called ternary relation.
 for n = 4 the relation R(x1, x2, x3, x4) is called quaternary relation.
 for n = 5 the relation R(x1, x2, x3, x4, x5) is called quinary relation.

6. Example:
X={1,2,3,4},
X*X={(1,1)(1,2)(1,3)(1,4)(2,1)(2,2)(2,3)(2,4),(3,1)(3,2)(3,
3)(3,4)(4,1)(4,2)(4,3)(4,4)}
Let the relation R be defined as
R={(x,y)/y=x+1,x,yЄX}
R={(1,2)(2,3)(3,4)
The relation matrix R is given by…..
1 2 3 4
1 0 1 0 0
2 0 0 1 0
3 0 0 0 1
4 0 0 0 0

7. Operations on relations:
Two relations R and S defined on X*Y and represented by relation
matrices following operations are supported by R and S
Union
R ∪ S (x , y) = max [ R (x , y) , S (x , y) ]
Intersection
R ∩ S (x , y) = min [ R(x , y) , S (x , y) ]
Complement
Ṝ(x , y)=1-R(x , y)

8. Composition of relations:
 Given R to be a relation on X,Y and S to be a relation on Y,Z then
R◌ S is a composition of elation on X,Z defined as
 R ◌S={(x,z)/(x,z) ЄX*Z, y ЄY such that (x,y) ЄR and (y,z) ЄS
 A common form of the composition relation is the max-min
composition

9. Max-Min Composition:
 The relation matrices of the relation R and S , the max-min
composition is defined as
T=R◌S
T(x , y)=max(min(R(x , y) , S(y , z)))

10. THANKYOU