Bijections.

Definition 2.4.1   A function $ f: \; A \longrightarrow B$ is a bijection (or is bijective) if it is both injective (v.s.  2.1 and surjective (v.s. 3.1), i.e. if every element of $ B$ has exactly one pre-image in $ A$ .

$ f:A \longrightarrow B$ is a bijection if, and only if:

$\displaystyle \forall y \in B, \quad \exists ! \; x \in A \; \vert \; y=f(x).$    

An easy corollary of Prop. 2.5 and Prop. 3.3 is the following:

Corollary 2.4.2   Let $ f:A \longrightarrow B$ and $ g: B \longrightarrow C$ be two applications. If $ f$ and $ g$ are bijective, then $ g o f : A \longrightarrow C$ is a bijection.

If $ A$ is a set, the identity of $ A$ is the application $ Id_A: \;A \longrightarrow A$ such that $ \forall x \in A, \; Id(x)=x$ .

Theorem 2.4.3   The application $ f:A \longrightarrow B$ is bijective if, and only if, there exists an application $ g:B \longrightarrow A$ such that $ gof=Id_A$ and $ fog=Id_B$ .

Figure 1: Bijections.
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The application $ g$ is called the inverse of $ f$ and is denoted $ g=f^{-1}$ . Of course $ f$ is the inverse of $ g$ and we denote $ g^{-1}=f$ .

Example 2.4.4   If $ f$ and $ g$ are mappings from $ \mathbb{R}$ to itself such that $ f(x)=2x-1$ and $ g(x)=\frac 12 (x+1)$ , then $ gof=Id_{\mathbb{R}}$ and $ fog=Id_{\mathbb{R}}$ .

Noah Dana-Picard 2007-12-28