Let $E$ be a set and $a$ and $b$ two elements of $E$. How do you write correctly the previous statment using formal language?
Let me explain where my question comes from.
When I define multiple variables, I oftenly write "Let $(a,b) \in E$ ..." by mimicking my teachers. The following examples are what makes me suspicious of this notation :
- $E$ = {$(x,y)$ | $x$ and $y$ gets some propertie(s)}
- $E$ = {$x$ | $x$ get some propertie(s)}
In the first example, $(a,b) \in E$ make sense to me because $(a,b)$ is an element of $E$ as a couple but $a \notin E$ and $b \notin E$, which is not as i wanted.
Now, the second example seems to have both $a$ and $b$ while $(a,b)$ is not. Looks good to me. Then why don't we write "Let $(a,b) \in E^{2}$ ..." ? I suppose it is some sort of convention, but I do not get why we use it since it could cause trouble when we use vectors : let's say $a$ and $b$ are real numbers, we usually write $v = \begin{pmatrix} a \\ b \end{pmatrix} \in \mathbb{R}^{2}$.
Let's resume.
Assuming $E$ is a set like the second example, do we have to write $(a,b) \in E^{2}$ or $(a,b) \in E$ ? In the lastest case, is it a convention or is there a reason i don't know why we do so ?
The notation you're referring to is not "incorrect" but lacks some rigour. Let me explain, for you first example where $E = \{(x,y):\text{ $x,y$ have some properties}\}$, the set is essentially a set of 2-tuples where the order matters. So if you were to say an element $x$ belonged to the set $E$ you could write it as $x \in E$ where $x = (x_1, x_2)$ where $x_1,x_2$ are the elements in the tuple. But strictly speaking $x$ is in $E$ and it is composed of a 2-tuple $(x_1, x_2)$.
For the second example, if you want to say two distinct elements are in $E$ you can just write $x,y \in E$. This is understood to mean that two distinct elements (unless you specify $x=y$) belong to the set $E$. If you wanted to do this for your first example, you could write $x,y \in E$ where $x=(x_1,x_2) \And y=(y_1,y_2)$.
Hope this helps :).