# Kenneth Rosen Edition 7th Exercise 2.3 Question 73 (Page No. 155)

45 views

Prove or disprove each of these statements about the floor and ceiling functions.

1. $\left \lceil \left \lfloor x \right \rfloor \right \rceil = \left \lfloor x \right \rfloor$ for all real number $x.$
2. $\left \lfloor 2x \right \rfloor = 2\left \lfloor x \right \rfloor$ whenever $x$ is a real number.
3. $\left \lceil x \right \rceil + \left \lceil y \right \rceil - \left \lceil x+y \right \rceil =0$ or $1$ whenever $x$ and $y$ are real numbers.
4. $\left \lceil xy \right \rceil = \left \lceil x \right \rceil \left \lceil y \right \rceil$ for all real numbers $x$ and $y.$
5. $\left \lceil x/2 \right \rceil = \left \lfloor x+1 / 2 \right \rfloor$ for all real numbers $x.$

## Related questions

1
38 views
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 $y.$
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 $x$ ... $f_{\sim A}= 1-f_{A} (x)$ $f_{A \oplus B}(x) = f_{A}(x) + f_{B}(x)- 2 f_{A}(x) f_{B}(x)$
Suppose that $f$ is an invertible function from $Y$ to $Z$ and $g$ is an invertible function from $X$ to $Y$. Show that the inverse of the composition $fog$ is given by $(fog)^{-1} = g^{-1} o f^{-1}.$