Composition
The composition for maps is just applying two maps in a row.
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Study Composition
Definition 1. If \(f:X \to Y\) and \(g : Y\to Z\), we may compose, or concatenate these maps: \[\begin{aligned} g \circ f : X &\to Z\\ x &\mapsto g(f(x)) \end{aligned}\] We call \(g \circ f\) the composition of the two functions.
Usually, \(g\circ f \neq f\circ g\), the latter does not even make sense, in general. \[X \to Y \to Z\]
Example 2.
\(f: \mathbb{R} \rightarrow \mathbb{R}\), \(x\mapsto x^2\); \(g:\mathbb{R} \rightarrow \mathbb{R}\), \(x\mapsto \sin(x)\) \[\begin{aligned} g\circ f: \mathbb{R} &\rightarrow \mathbb{R} \\ x &\mapsto \sin(x^2) \\ f\circ g: \mathbb{R} &\rightarrow \mathbb{R} \\ x &\mapsto (\sin(x))^2 \end{aligned}\]
Let \(X\) be a set. Then \(\operatorname{id}_X: X\rightarrow X\) with \(x\mapsto x\) is called the identity map. If there is no confusion, one usually writes \(\operatorname{id}\) instead of \(\operatorname{id}_X\). Let \(f: X\rightarrow X\) be a function. Then \[f\circ \operatorname{id}=f=\operatorname{id}\circ f.\]
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