Cartesian product of two real sets I've two sets, Here: $A=(0,5]$ and $B=[2,4]$
The following product is right or wrong?  $A\times B=[(0,2);(0,4);(5,2);(5,4)]$
 A: The Cartesian Product of two sets $A$ and $B$ is defined as
$$
\{(x,y): x \in A \text{ and } y \in B\}
$$
(0,2) and (0,4) would not be in the Cartesian product because $0$ is not in $A$.
A: It is wrong as $A\times B $ here is an infinite set but yours' contains only 4 elements. You may proceed as follows to get the Cartesian product:The Cartesian product of $A\times B=\{(x, y) : x\in (0,5], y\in [2,4]\}\tag{1}$ 
Let coordinates of $A, B, C, D $ are $(0,2),(5,2),(5,4), (0,4)$ respectively. So geometrically, in this case, $A\times B$ represents rectangle $ABCD$ without side $AD$ as $(0,5]$ is open at $0$.
A: The ";" notation, as I'm sure you've gathered from the comments, is not typical; I certainly don't recognize it. To understand what's going on here, I think it's useful to briefly revisit the relevant notations.
"$[a,b]$" means "all numbers from $a$ to $b$, including $a$ and $b$". For example, $[2,4]$ includes $2$, $4$, $3$, $2.8$, $\pi$, and any other number you can think of that's in between $2$ and $4$.
"$(a,b]$" means "all numbers from $a$ to $b$, including $b$ but not $a$". For example, $(0,5]$ includes $1$, $5$, $4$, $\pi$, $\sqrt{2}$, and any other number you can think of that's in between $0$ and $5$ - except $0$ itself.
In general, "$\{p : $ (fact about $p$) $\}$" is the collection of all things that could be named "$p$" for which that fact is true. For example, $\{x : 2 \leq x \leq 4\}$ would be "all numbers $x$ so that $x$ is between $2$ and $4$ inclusive" - in other words, it would be another way to describe $[2,4]$.
The notation you're using, $[(0,2); (0,4); (5,2); (5,4);]$ isn't one of these. If you mean to write "the set containing these four points and nothing else", the correct notation would be $\{(0,2), (0,4), (5,2), (5,4)\}$; note the use of the $\{\}$ instead of square brackets, and commas instead of semicolons.
Regardless, that's not the correct Cartesian product. Remember that the Cartesian product of two sets $X$ and $Y$ is defined by
$$X \times Y = \{(a,b) : a \in X \text{ and } b \in Y\}$$
meaning that something is in $X \times Y$ if it is a pair whose first component is in $X$ and whose second component is in $Y$. In this case, we want:
$$A \times B = \{(a,b) : a \in A \text{ and } b \in B\}$$
Using our definitions of $A$ and $B$, we can rewrite this as:
$$A \times B = \{(a,b) : 0 < a \leq 5 \text{ and } 2 \leq b \leq 4\}$$
That's about as good of a presentation that we can manage - there's no better way to write this that I'm aware of. But we can analyze it a little, to help us make sense of it. For example:
Is (1, 3) in $A \times B$? Well, $0 < 1 \leq 5$, and $2 \leq 3 \leq 4$, so yes, it is.
Is (2,7) in $A \times B$? $0 < 2 \leq 5$, but it isn't correct to say that $2 \leq 7 \leq 4$, so no, it isn't.
Is $(e,\pi)$ in $A \times B$? Since $0 < e \leq 5$ and $2 \leq \pi \leq 4$, yes, it is.
By applying this kind of thinking more and more, a good general picture emerges: $A \times B$ is the rectangle with $x$-coordinates between $0$ and $5$ and $y$-coordinates between $2$ and $4$, except that all of the points with $x$-coordinate $0$ have been removed.
