6.1 Floor and ceiling

The floor of a real number $r$ is the largest integer that is less than or equal to $r$ . The ceiling of a real number $r$ is the least integer that is greater than or equal to $r$ . The floor of $r$ is usually represented as $\lfloor r \rfloor$ , whereas the ceiling of $r$ is usually represented as $\lceil r \rceil$ . Because of their similarities, we will only consider the floor function here.

The floor function is a function because every real number $r$ has $\lfloor r \rfloor$ defined. In other words, $\forall x \in \mathbb{R}: \exists y \in \mathbb{I}$ .

The floor function is not an injection because $0 = \lfloor 0.5 \rfloor = \lfloor 0.2 \rfloor$ .

The floor function is a surjection because $\forall y \in \mathbb{I}: (y, y) \in L$ .