Composition

Composition of Relations

Let $R:A\rightarrow{B}$ and $S:B\rightarrow{C}$ are 2 relations. Composition can be defined when $\text{ran}(R)=\text{preran}(S)$.

Say $\text{ran}(R)=\text{preran}(S)=D$. Composition of the 2 relations is written as:

Identity Relation

From the properties of the inverse relation, $R\circ R^{-1}, R^{-1}\circ R$ are both defined always. This relation is called the identity relation and denoted by $I$.

Composition of Functions

Let $f:A\rightarrow{B}$ and $g:B\rightarrow{C}$ be 2 functions where $f$ is onto.

The notation $g\circ f$ can be written as $g(f(x))$.