# Max Min composition for crisp relation

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 = { (x_{1}, y_{1}), (x_{1}, y_{3}), (x_{2}, y_{4}) } and S = { (y_{1}, z_{2}), (y_{3}, z_{2}) } . 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

χ_{T}(x_{1}, z_{1}) = max(min( χ_{R}(x_{1}, y_{1}), χ_{S}(y_{1}, z_{1})), min( χ_{R}(x_{1}, y_{2}), χ_{S}(y_{2}, z_{1})), min( χ_{R}(x_{1}, y_{3}), χ_{S}(y_{3}, z_{1})), min( χ_{R}(x_{1}, y_{4}), χ_{S}(y_{4}, z_{1}))

χ_{T}(x_{1}, z_{1}) = max(min(1, 0),min(0, 0),min(1, 0),min(0, 0))

χ_{T}(x_{1}, z_{1}) = max(0,0,0,0) = 0

Similarly,

χ_{T}(x_{1}, z_{2}) = max(min(1, 1),min(0, 0),min(1, 1),min(0, 0)) = max(1, 0, 1, 0) = 1

χ_{T}(x_{2}, z_{1}) = max(min(0, 0),min(0, 0),min(0, 0),min(1, 0)) = max(0, 0, 0, 0) = 0

χ_{T}(x_{2}, z_{2}) = max(min(0, 1),min(0, 0),min(0, 1),min(1, 0)) = max(0, 0, 0, 0) = 0

χ_{T}(x_{3}, z_{1}) = max(min(0, 0),min(0, 0),min(0, 0),min(0, 0)) = max(0, 0, 0, 0) = 0

χ_{T}(x_{3}, z_{2}) = 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,

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## 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,

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

χ_{T}(1, 1) = max(min(0, 0), min(1, 0), min(0, 0)) = max(0, 0, 0) = 0

χ_{T}(1, 3) = max(min(0, 1), min(1, 0), min(0, 0)) = max(0, 0, 0) = 0

χ_{T}(1, 5) = max(min(0, 1), min(1, 1), min(0, 0)) = max(0, 1, 0) = 1

χ_{T}(3, 1) = max(min(0, 0), min(0, 0), min(1, 0)) = max(0, 0, 0) = 0

χ_{T}(3, 3) = max(min(0, 1), min(0, 0), min(1, 0)) = max(0, 0, 0) = 0

χ_{T}(3, 5) = max(min(0, 1), min(0, 1), min(1, 0)) = max(0, 0, 0) = 0

χ_{T}(5, 1) = max(min(0, 0), min(0, 0), min(0, 0)) = max(0, 0, 0) = 0

χ_{T}(5, 3) = max(min(0, 1), min(0, 0), min(1, 0)) = max(0, 0, 0) = 0

χ_{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,

## Test your knowledge:

For the given relations,

Find max-min composition.

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

This is really helpful

Thank you Urmi

Very informative and helpful, nicely explained

Thank you very much Preeti for your words

Ahh… What a simple and easy explanation with examples!

Thank for your words Shivang

simply explained.

Thanks dear

Very Helpful

. Thanks.Noted with thanks !

Everything explained so nicely.

Thanks for interaction

Max Min composition Explained wonderfully

Noted with thanks

please check XT(5,3) and XT(5,5) in the order pairs of the min terms