Negation of floor
Dependencies: None
Let $x \in \mathbb{R}$. Then $\lfloor -x \rfloor = -\lceil x \rceil$.
Proof
By the definition of floor and ceil, we get that $\lfloor x \rfloor = n \iff x \in [n, n+1)$ and $\lceil x \rceil = n \iff x \in (n-1, n]$. \[ \lfloor -x \rfloor = n \iff -x \in [n, n+1) \iff x \in (-n-1, -n] \iff \lceil x \rceil = -n \]
Dependency for: None
Info:
- Depth: 0
- Number of transitive dependencies: 0