Mapping

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:

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