Consider the mapping \(F : \mathbb{R} \to \mathbb{Z} \times [0,1)\) defined as

\[F(x) = \left( \lfloor x \rfloor, \{x\} \right)\]

for all \(x \in \mathbb{R}\). Is this mapping injective? Is this mapping surjective?

**Notations**:

\( \lfloor \cdot \rfloor \) denotes the floor function.

\( \{ \cdot \} \) denotes the fractional part function.

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