Prove or disprove each of these statements about the floor and ceiling functions. $\left \lfloor \left \lceil x \right \rceil \right \rfloor = \left \lceil x \right \rceil$ for all real numbers $x.$ ... $x$ and $y.$

Prove or disprove each of these statements about the floor and ceiling functions. $\left \lceil \left \lfloor x \right \rfloor \right \rceil = \left \lfloor x \right \rfloor$ for all real number $x.$ $\left \lfloor 2x \right \rfloor = 2\left \lfloor x \right \rfloor$ whenever $x$ is a ... $x$ and $y.$ $\left \lceil x/2 \right \rceil = \left \lfloor x+1 / 2 \right \rfloor$ for all real numbers $x.$

Suppose that $f$ is a function from $A$ to $B$, where $A$ and $B$ are finite sets with $|A|=|B|$. Show that $f$ is one-to-one if and only if it is onto.

Let $S$ be a subset of a universal set $U$. The characteristic function $f_{s}$ of $S$ is the function from $U$ to the set $\left \{ 0,1 \right \}$ such that $f_{S}(x)=1$ if $x$ belongs to $S$ and $f_S(x)=0$ if $x$ does not belong to $S$. Let $A$ and $B$ be sets. Show that for all $x$ ... $f_{\sim A}= 1-f_{A} (x)$ $f_{A \oplus B}(x) = f_{A}(x) + f_{B}(x)- 2 f_{A}(x) f_{B}(x) $