notation
I am confused by the notation comma.
I know that the comma means 'AND' in Set theory as gate ($a \land b=a$ AND $b$),
But we write solution of equation as $x=1$, $2$ (the equation: $x^2-3x+2=0$)
The question is whether $x=1,2$ is WRONG?
$x=1,2$ $\iff$ $x=1$ AND $x=2$, so rewritten as $x=1$ or $x=2$.
But in so many books, it is written as $x=1$, $2$.
I am confused…..
(PS. I think the comma of $x=1$, $2$ is only notation of classification…? Is it ok?)
$$ '\{ (1,2,3,\dots, n); n\in \mathbb{N}\}' $$
is the same thing as
$$ \{(1), (1,2),\ldots \} $$
Check out:
I don't think we have $\Omega = \{(0,0,...) \cup (1,1,...)\}$
In the area of logic, $\longleftrightarrow$ is usually used for "if and only if" instead of $\iff$ (because who wants to bother drawing that second line all the time).
Otherwise when dealing with functions, $\longleftrightarrow$ might also be used to denote a bijective function. So $f \colon A \leftrightarrow B$ is a bijection between $A$ and $B$. Or you could similarly write $$ A \overset{f}{\longleftrightarrow} B $$
In regards to what was likely meant in the video that you saw, the following is true:
For a given value of $x$, one has $\sum\limits_{n=1}^\infty \frac{1}{n^x}$ converges if and only if $\int\limits_{1}^\infty \frac{1}{t^x}dt$ converges.
Best Answer
First of all you should know, that there are more countries where the comma is a decimal separator than there are point-separator-countries. (For example: I live in Austria in Europe, and we use the comma as decimal separator.) The international standard since about 100 years is to use a point as decimal separator (before that time the comma was the international decimal separator).
blue: decimal separator is a point ($\pi = 3.14$)
green: decimal separator is a comma ($\pi = 3$,$14$)
red: decimal separator is a momayyez ($\pi = 3٫14$)
other colors: two or all three of the above standards are in use
In Countries where the comma is not used as decimal separator (not-green countries in the picture), it is used as list-separator, for example when you want to list the elements of a set. This is also used internationally:
In countries where the decimal-separator is a comma (green countries), the semicolon is used als list-separator:
Often a mathematical problem has more than one solutions (for example $x^2-5x+6$). So the solution is a set of numbers:
But a shorter way to write the same fact is this: