2. Fuzzy Relations Objectives: Crisp and Fuzzy Relations Projections and Cylindrical Extensions* Extension Principle Compositions of Fuzzy Relations Ming-Feng Yeh

Cartesian Product: crisp Let A and B be two crisp subsets in X and Y, respectively. The Cartesian product of A and B, denoted by AB, is defined by Let X={0, 1}, Y={a,b,c}. If A=X and B=Y, then AB={(0,a), (0,b), (0,c), (1,a), (1,b), (1,c)} BA={(a,0), (b,0), (c,0), (a,1), (b,1), (c,1)} Ming-Feng Yeh

Fuzzy Relationships A fuzzy relationship over the pair X, Y is defined as a fuzzy subset of the Cartesian product XY. If X={0, 1}, Y={a,b,c}, then A = {0.1/(0,a), 0.6/(0,b), 0.8/(0,c), 0.3/(1,a), 0.5/(1,b), 0.7/(1,c)} is a fuzzy relationship over the space XY. Ming-Feng Yeh

Cartesian Product: fuzzy Let A and B be fuzzy sets in X and Y, respectively. The Cartesian product of A and B, denoted by AB, is a fuzzy set in the product space XY with the membership function: Assume X={0,1} and Y={a,b,c} Let A=1.0/0 + 0.6/1, B=0.2/a + 0.5/b+ 0.8/c. Then AB is a fuzzy relationship over XY. Ming-Feng Yeh

Cylindrical Extension* Assume X and Y are two crisp sets and let A be a fuzzy subset of X. The cylindrical extension of A to XY, denoted by , is a fuzzy relationship on XY. Assume X={a,b,c} and Y={1,2}. Let A={1/a, 0.6/b, 0.3/c}. Then the cylindrical extension of A to XY is {1/(a,1), 1/(a,2), 0.6/(b,1), 0.6/(b,2), 0.3/(c,1), 0.3/(c,2)} Ming-Feng Yeh

Cylindrical Extension* A(x) x y x Ming-Feng Yeh

Projection* Assume A is a fuzzy relationship on XY. The projection of A onto X is a fuzzy subset A of X, denoted by A=Projx A, Assume X = {a,b,c} and Y = {1,2}. Let A={1/(a,1), 0.6/(a,2), 0.8/(b,1), 0.6/(b,2), 0.3/(c,1), 0.5/(c,2)}. Then Projx A = {1/a, 0.6/b, 0.5/c}. Projy A = {0.8/1, 0.6/2}. Ming-Feng Yeh

Projection* Ming-Feng Yeh

Extension Principle Assume X and Y are two crisp sets and let f be a mapping form X into Y, f: XY, such that xX, f(x) = y Y. Assume A is a fuzzy subset of X, using the extension principle, we can define f(A) as a fuzzy subset of Y such that Denote B = f(A), then B is a fuzzy subset of Y such that for each y Y Ming-Feng Yeh

Example 2-3 Assume X = {1, 2, 3} and Y = {a, b, c, d, e}. Let f be defined by f(1) = a, f(2) = e, f(3) = b. Let A = {1.0/1, 0.3/2, 0.7/3} be a fuzzy subset, then B = f(A) = {1.0/a, 0.3/e, 0.7/b}. Let A = 0.1/2 + 0.4/1 + 0.8/0 + 0.9/1 + 0.3/2 and f(x) = x2 3. Then B = 0.1/1 + 0.4/2 + 0.8/3 + 0.9/2 + 0.3/1 = 0.8/3 + (0.40.9)/2 + (0.10.3)/1 = 0.8/3 + 0.9/2 + 0.3/1 Ming-Feng Yeh

Binary Fuzzy Relations Let X and Y be two universes of discourse. Then is a binary fuzzy relation in XY. Examples of binary fuzzy relation: y is much greater than x. (x and y are numbers) x is close to y. (x and y are numbers) x depends on y. (x and y are events) x and y look alike. (x and y are persons, objects, etc.) If x is large, then y is small. (x is an observed reading and y is a corresponding action) Ming-Feng Yeh

Max-min Composition Let R1 and R2 be two fuzzy relations defined on XY and YZ, respectively. The max-min composition of R1 and R2 is a fuzzy set defined by Max-min product: the calculation of is almost the same as matrix multiplication, except that  and  are replaced by  and , respectively. Ming-Feng Yeh

Max-product Composition Let R1 and R2 be two fuzzy relations defined on XY and YZ, respectively. The max-product composition of R1 and R2 is a fuzzy set defined by Ming-Feng Yeh

Example 2-3 R1 = “x is relevant to y”, R2 = “y is relevant to z”, X = {1,2,3}, Y={,,,} and Z={a,b}. Max-min composition: Max-product composition: Ming-Feng Yeh