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

R âˆ˜ S = {(x, z) | (x, y) âˆˆ R, and (y, z) âˆˆ S, âˆ€y âˆˆ 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:

Matrix representation:

The final composition of relation would look something like this,

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

Ï‡T(x1, z1) = max(min( Ï‡R(x1, y1), Ï‡S(y1, z1)), min( Ï‡R(x1, y2), Ï‡S(y2, z1)), min( Ï‡R(x1, y3), Ï‡S(y3, z1)), min( Ï‡R(x1, y4), Ï‡S(y4, z1))

Ï‡T(x1, z1) = max(min(1, 0),min(0, 0),min(1, 0),min(0, 0))

Ï‡T(x1, z1) = max(0,0,0,0) = 0

Similarly,

Ï‡T(x1, z2) = max(min(1, 1),min(0, 0),min(1, 1),min(0, 0)) = max(1, 0, 1, 0) = 1

Ï‡T(x2, z1) = max(min(0, 0),min(0, 0),min(0, 0),min(1, 0)) = max(0, 0, 0, 0) = 0

Ï‡T(x2, z2) = max(min(0, 1),min(0, 0),min(0, 1),min(1, 0)) = max(0, 0, 0, 0) = 0

Ï‡T(x3, z1) = max(min(0, 0),min(0, 0),min(0, 0),min(0, 0)) = max(0, 0, 0, 0) = 0

Ï‡T(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,

## 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:

XÃ—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,

T= R âˆ˜ S = {(x, z) | (x, y) âˆˆ R, and (y, z) âˆˆ S, âˆ€y âˆˆ Y}

Ï‡T(1, 1) = max(min(0, 0), min(1, 0), min(0, 0)) = max(0, 0, 0) = 0

Ï‡T(1, 3) = max(min(0, 1), min(1, 0), min(0, 0)) = max(0, 0, 0) = 0

Ï‡T(1, 5) = max(min(0, 1), min(1, 1), min(0, 0)) = max(0, 1, 0) = 1

Ï‡T(3, 1) = max(min(0, 0), min(0, 0), min(1, 0)) = max(0, 0, 0) = 0

Ï‡T(3, 3) = max(min(0, 1), min(0, 0), min(1, 0)) = max(0, 0, 0) = 0

Ï‡T(3, 5) = max(min(0, 1), min(0, 1), min(1, 0)) = max(0, 0, 0) = 0

Ï‡T(5, 1) = max(min(0, 0), min(0, 0), min(0, 0)) = max(0, 0, 0) = 0

Ï‡T(5, 3) = max(min(0, 1), min(0, 0), min(1, 0)) = max(0, 0, 0) = 0

Ï‡T(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,