3. Some useful functions

Useful functions

Floor function

Floor function

The floor function, $\lfloor \rfloor :\mathbb{R}\to \mathbb{Z}$, assigns any real number $x$ to the largest integer that is less than or equal to $x$.

Examples:

• $\lfloor \frac{1}{2}\rfloor =0$
• $\lfloor -\frac{3}{2}\rfloor =-2$
• $\lfloor 2.3\rfloor =3$

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Ceiling function

Ceiling function

The ceiling function, $\lceil \rceil :\mathbb{R}\to \mathbb{Z}$, assigns any real number $x$ to the smallest integer that is greater than or equal to $x$.

Examples:

• $\lceil \frac{1}{3}\rceil =1$
• $\lceil -\frac{5}{4}\rceil =-1$

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Properties of Floor and Ceil

Here are some simple properties of both floor and ceil functions.

$$\begin{array}{c}x-1<\lfloor x\rfloor \le x\le \lceil x\rceil <x+1\\ \lfloor -x\rfloor =-\lceil x\rceil \\ \lceil -x\rceil =-\lfloor x\rfloor \end{array}$$

Functions with multiple arguments

If the domain of a function $f$ is a Cartesian product $A_1\times ...\times A_n$, we can say that:

• $f$ has arity $n$

  Arity

  Arity is the number of arguments or operands taken by a function.

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• $f$ is an $n$-ary function
• $f$ has $n$ arguments

In this case, for each $n$-tuple $(x_1,...,x_n)\in A_1\times ...\times A_n$, $\boxed{f(x_1,...,x_n)}$ denotes the value of $f$.

Binary function

A binary function $f$ has two arguments.

For binary functions, we may write either of:

• $f(x,y)=z$ $+(4,5)=9$
• $xfy=z$ $4+5=9$

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