Max Min composition is one way of computing interaction between variables of different relations. Composition of relation R and S is denoted as R ∘ S. Mathematically, it is defined as,

R ∘ S = {(x, z) | (x, y) ∈ R, and (y, z) ∈ S, ∀y ∈ Y}

Composition of relation is computed in two different ways:

Although, for crisp relation both are identical, for fuzzy relations, results of max-min composition and max-product composition would be different.

Example 1: Max min composition

We will try to understand max min composition with multiple examples. Let R = { (x1, y1), (x1, y3), (x2, y4) } and S = { (y1, z2), (y3, z2) } . let us find Max-Min composition of these relations.

As we know, representation of crisp relation could take multiple forms. The above relation R and S we can represent as,

Sagittal representation:

sagittal representation of crisp realtion

Matrix representation:

The final composition of relation would look something like this,

Composition of relations R and S
Composition of relations R and S

Let us see how to fill the cells of composition matrix T

χT(x1, z1) = max(min( χR(x1, y1), χS(y1, z1)), min( χR(x1, y2), χS(y2, z1)), min( χR(x1, y3), χS(y3, z1)), min( χR(x1, y4), χS(y4, z1))

χT(x1, z1) = max(min(1, 0),min(0, 0),min(1, 0),min(0, 0))

χT(x1, z1) = max(0,0,0,0) = 0

Similarly,

χT(x1, z2) = max(min(1, 1),min(0, 0),min(1, 1),min(0, 0)) = max(1, 0, 1, 0) = 1

χT(x2, z1) = max(min(0, 0),min(0, 0),min(0, 0),min(1, 0)) = max(0, 0, 0, 0) = 0

χT(x2, z2) = max(min(0, 1),min(0, 0),min(0, 1),min(1, 0)) = max(0, 0, 0, 0) = 0

χT(x3, z1) = max(min(0, 0),min(0, 0),min(0, 0),min(0, 0)) = max(0, 0, 0, 0) = 0

χT(x3, z2) = max(min(0, 1),min(0, 0),min(0, 1),min(0, 0)) = max(0, 0, 0, 0) = 0

Thus, the composition of relation R and S would be,

min max composition of relation R and S
Composition of relation R and S

Watch on YouTube:

max min composition on youtube

Example 2: Max min composition

Given X={1, 3, 5}, Y={1, 3, 5},

R = { (x, y) | y = x + 2 } = { (1, 3), (3, 5) }

S = { (x, y) | x < y } = { (1, 3), (1, 5), (3, 5) }

Find Max-Min composition of relations R and S

Solution:

X×Y={(1, 1), (1, 3), (1, 5),(3, 1), (3, 3), (3, 5),(5, 1), (5, 3), (5, 5)}

R = { (x, y) | y = x + 2 } = { (1, 3), (3, 5) }

S = { (x, y) | x < y } = { (1, 3), (1, 5), (3, 5) }

In matrix form, we can represent relation R and S as,

relation r
relation s

T= R ∘ S = {(x, z) | (x, y) ∈ R, and (y, z) ∈ S, ∀y ∈ Y}

χT(1, 1) = max(min(0, 0), min(1, 0), min(0, 0)) = max(0, 0, 0) = 0

χT(1, 3) = max(min(0, 1), min(1, 0), min(0, 0)) = max(0, 0, 0) = 0

χT(1, 5) = max(min(0, 1), min(1, 1), min(0, 0)) = max(0, 1, 0) = 1

χT(3, 1) = max(min(0, 0), min(0, 0), min(1, 0)) = max(0, 0, 0) = 0

χT(3, 3) = max(min(0, 1), min(0, 0), min(1, 0)) = max(0, 0, 0) = 0

χT(3, 5) = max(min(0, 1), min(0, 1), min(1, 0)) = max(0, 0, 0) = 0

χT(5, 1) = max(min(0, 0), min(0, 0), min(0, 0)) = max(0, 0, 0) = 0

χT(5, 3) = max(min(0, 1), min(0, 0), min(1, 0)) = max(0, 0, 0) = 0

χT(5, 5) = max(min(0, 1), min(1, 0), min(0, 0)) = max(0, 0, 0) = 0

Thus, the composition of relation R and S would be,

Composition of relations R and S

Test your knowledge:

For the given relations,

matrix-a
matrix-b

Find max-min composition.

Please post your answer / query / feedback in comment section below !