6.2 Frequently asked questions

Here you will find an overview of most frequent questions, problems and mistakes which students meet especially at the beginning of the course and which can already be discussed now.

6.2.1 We did/called/denoted it differently at high school

It is possible and it is also quite all right. However, you cannot expect that all people around the world will comply with the approach you were taught at high school. Different people may use different conventions and may have very good reasons for doing so.

If you use some materials to study it is a good idea to first make yourself familiar with the language which is used in the text. For example, the BIE-ZMA course material starts with a list of symbols and ends with a list of names which helps readers to find definitions they need quickly.

We can demonstrate this situation on the names used in connection with monotonous functions and sequences (increasing, strictly increasing, non-increasing, etc.). There are a lot of different nomenclatures. However, we will only use one set of names to avoid misunderstanding. See lectures and lecture notes.

This remark does not only apply to mathematics but it is generally valid.

6.2.2 Inclusion

Not only in BIE-ZMA, the symbol $\subset$ is used to denote inclusion and we do not distinguish between a subset and a proper subset. I.e., the inclusion $A \subset B$ holds if and only if every element of the set $A$ is also an element of the set $B$. In particular, for any set $A$ we have that $A \subset A$. We can make do with this one notion without a problem throughout the whole course.

Texts which do distinguish between subsets and proper subsets usually use special symbols $A \subseteq B$ and $A \subsetneq B$ for that end.

6.2.3 Domains of trigonometric functions

The functions $\sin$, $\cos$, $\mathrm{tg}$ are not injective and therefore they do not have inverses. However, we can restrict them to sets on which they are injective and then construct inverse functions on such sets. There are infinitely many ways of rectricting these functions in such a way. The standard choice is the following:

\begin{equation*} \begin{aligned} \arcsin &= \left( \sin \Big|_{\langle -\pi/2, \pi/2 \rangle} \right)^{-1}, \\ \arccos &= \left( \cos \Big|_{\langle 0, \pi \rangle} \right)^{-1}, \\ \mathrm{arctg} &= \left( \mathrm{tg} \Big|_{( -\pi/2, \pi/2 )} \right)^{-1}. \end{aligned} \end{equation*}

Hence we have that

\begin{equation*} \begin{aligned} D_{\arcsin} &= \langle -1, 1 \rangle, & H_{\arcsin} &= \langle -\pi/2, \pi/2 \rangle, \\ D_{\arccos} &= \langle -1, 1 \rangle, & H_{\arccos} &= \langle 0, \pi \rangle, \\ D_{\mathrm{arctg}} &= \mathbb{R}, & H_{\mathrm{arctg}} &= (-\pi/2, \pi/2). \end{aligned} \end{equation*}

6.2.4 Zero to the power of zero

The algebraic expression $0^0$ is defined as $1$. Zero in the exponent denotes an empty multiplication (there are no numbers to be multiplied) and so the result is the identity element for multiplication which is $1$. Similarly, the empty sum evaluates to $0$, the identity element for addition.

6.2.5 Necessary condition, direction of an implication

If an implication $A \Rightarrow B$ is true then we often call $B$ a necessary condition for $A$. If $B$ is not true then $A$ cannot be true either (because if $A$ were true then so would be $B$).