CSCE 222 Lecture 11

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Functions

$f$ is surjective iff range($f$) = $B$

$f$ is injective iff $f(a)=f(b)$ implies $a=b$

thus $a\neq b$ implies $f(a)\neq f(b)$

$f$ is bijective iff surjective and injective

Example

$f:\mathbb {Z} \rightarrow \mathbb {Z} ,f(x)=x+1$ is bijective

$f(x)=x+1=y+1=f(y)$ implies $x=y$, so $f$ is injective

$\forall y\in \mathbb {Z} \ \exists x\in \mathbb {Z} :y=f(x);x=y-1$, so $f$ is surjective

Inverse Functions

$f:A\rightarrow B$

$f^{-1}:B\rightarrow A$ is a function that reverses the direction of mappings so $f(a)=b\equiv f^{-1}(b)=a$

Floor

floor ($\lfloor x\rfloor$)

maps and argument $x$ to the largest integer that doesn't exceed $x$

EX: $\lfloor 1/2\rfloor =0$

Properties

given $x\in \mathbb {R}$ and $n\in \mathbb {Z}$,

$\lfloor x\rfloor =n\quad n\leq x<n+1\ {\mbox{and}}\ x-1<n\leq x$

Addition and subtraction of integers:

$\lfloor x+m\rfloor =n+m\quad (m\in \mathbb {Z} )$

Multiplication:

$\lfloor mx\rfloor =\lfloor x\rfloor +\left\lfloor x+{\frac {1}{m}}\right\rfloor +\ldots +\left\lfloor x+{\frac {m-1}{m}}\right\rfloor$

Ceiling

ceil ($\lceil x\rceil$)

maps an argument $x$ to the smallest integer $\geq x$

EX: $\lceil -1.1\rceil =-1$

Properties

given $x\in \mathbb {R}$ and $n\in \mathbb {Z}$,

$\lceil x\rceil =n\quad n-1<x\leq n\quad x\leq n<x+1$

Note: $\lceil x+y\rceil \neq \lceil x\rceil +\lceil y\rceil$

Sum Notation

$\sum _{j=m}^{n}a_{j}$

Important Sums to Know

$\sum _{j=1}^{n}k={\frac {n(n+1)}{2}}$