Crisp relations are subsets of Cartesian products between sets. The Cartesian product of two sets A and B is the set of all ordered pairs where the first element is in A and the second is in B. Relations can be binary, ternary, quaternary, etc. depending on the number of sets involved. Operations like union, intersection, and complement can be performed on relations. Composition of relations R and S is defined as the set of all ordered pairs where the first element is related to the second by R and the second is related to the third by S. Max-min composition defines the composition relation matrix T as taking the max of the min of the relation matrices of R and S.

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