In the mathematics of binary relations, the formation of a new binary relation R∘S from two given binary relations R and S is known as the composition of relations. The composition of relations is known as relative multiplication in the calculus of relations, and the result is called a relative product.

What is Composition of Relations

Let A, B, and C be three sets and let R be a relation from A to B and S be a relation from Q to R.

The composition of R and S: S∘R is a binary relation from A to C, if and only if there exists a b∈B such that aRb and bSc.

Formally, the composition of S, R can be written as:

a(R◦S)c if for some b ∈ B we have aRb and bSc.
R ◦ S = {(a, c)| there exists b ∈ B for which (a, b) ∈ R and (b, c) ∈ S},

Get the tech career you deserve, faster!

Connect with our expert counsellors to understand how to hack your way to success

User rating 4.7/5

1:1 doubt support

95% placement record

Akash Pal

Senior Software Engineer

326% Hike After Job Bootcamp

Himanshu Gusain

Programmer Analyst

32 LPA After Job Bootcamp

After Job Bootcamp

Composition of Relations

The composition of binary relations is associative: R∘(S∘T) = (R∘S)∘T.

The composition of binary relations is not commutative: R∘S ≠ S∘R.

The composition of relations R and S, R◦S is often thought of as their multiplication andis written as RS.

Here is an example of the composition of relations:

This is because the composition of relations is associative, meaning that the arrangement of the relations does not matter.

Powers of Binary Relations

When a relation is defined on a set, it can always be composed of itself. As a result, we may have

R◦R is denoted by R^{2}. Similarly, R^{3} = R^{2}◦R = R◦R◦R, and so on.

Hence R^{n} is defined for all n > 0.

Composition of Relations in Matrix Form

Let the relations R and S are defined by their matrices M_{R} and M_{S}. Then, the composition of relations S∘R = RS is calculated by the matrix product of M_{R} and M_{S}:

M_{S∘R} = M_{RS} = M_{R} X M_{S}.

For example,

Let P = {1, 3, 4}. Consider the relation R and S on P defined by

What is the symbol for the composition of a relation?

A small circle () is positioned between the two relation names to represent the composition of a relation, such as R∘S, which symbolises the composition of relation R followed by relation S.

What is composition in set theory?

The composition of a function is a step-by-step process in mathematics. The functions f: A→ B & g: B→ C, for example, can be combined to generate a function that maps x in A to g(f(x)) in C. There are no empty sets in any of the sets. (g o f) (x) = g (f(x) denotes a composite function. The phrase g o f read as "g of f."

How do you solve composition of relations?

The composition of two relations is calculated by matrix product. Let the relations R and S are defined by their matrices M_{R} and M_{S}. Then, the composition of relations S∘R = RS is calculated by the matrix product of M_{R} and M_{S}. _{ }

What is a relation composition with itself?

Relation composition with itself, denoted as R∘R, refers to combining a relation R with itself. This operation yields a new relation consisting of pairs derived from composing elements of R with elements of R.

Conclusion

In this article, we learned to calculate the composition of relations. We learned that in the given three sets, A, B, and C and relations R be a relation from A to B and S be a relation from Q to R.

The composition of R and S: S∘R is a binary relation from A to C, if and only if there exists a b∈B such that aRb and bSc.

We also learned about the properties of the composition of relations. We then calculated the composition of relations in matrix form with the help of an example.