Given two fuzzy sets: A ={(a1,0.3),(a2,0.6),(a3,0.8)} , B={(b1,0.4),(b2,0.5)} find R=(B×A) ?

The Cartesian product of two fuzzy sets A and B, denoted as B×A, is a fuzzy relation R from B to A. It is defined as a set of ordered pairs, where each pair consists of an element from B and an element from A, along with a membership function that assigns to each pair a degree of membership ranging from 0 to 1.

The degree of membership for a pair (b, a) in the relation R is defined as the minimum of the degrees of membership of b in B and a in A. This is because the relation R is defined in terms of the logical conjunction (AND operation) of the memberships of b in B and a in A, and the conjunction is modeled by the minimum operation in fuzzy logic.

So, to find the fuzzy relation R = B×A for the given fuzzy sets A and B, we need to form all possible pairs of elements from B and A, and for each pair, we calculate the degree of membership as the minimum of the degrees of membership of the elements in their respective sets.

Here are the steps:

1. Form all possible pairs (b, a), where b is an element of B and a is an element of A:

   {(b1, a1), (b1, a2), (b1, a3), (b2, a1), (b2, a2), (b2, a3)}

2. For each pair, calculate the degree of membership as the minimum of the degrees of membership of the elements in B and A:

   R = {((b1, a1), min(0.4, 0.3)), ((b1, a2), min(0.4, 0.6)), ((b1, a3), min(0.4, 0.8)), ((b2, a1), min(0.5, 0.3)), ((b2, a2), min(0.5, 0.6)), ((b2, a3), min(0.5, 0.8))}

3. Simplify the degrees of membership:

   R = {((b1, a1), 0.3), ((b1, a2), 0.4), ((b1, a3), 0.4), ((b2, a1), 0.3), ((b2, a2), 0.5), ((b2, a3), 0.5)}

So, the fuzzy relation R = B×A is:

R = {((b1, a1), 0.3), ((b1, a2), 0.4), ((b1, a3), 0.4), ((b2, a1), 0.3), ((b2, a2), 0.5), ((b2, a3), 0.5)}