TITLE: Set theoretic definition of cartesian product of two sets
QUESTION [6 upvotes]: I am trying to study mathematics rigorously.It is mentioned in my book in the chapter of set theory that the symbol "$:$" stands for the phrase "such that". 
Then in the next chapter of relations, the cartesian product of two sets $A$ and $B$ is defined as follows:
If $A$ and $B$ be two sets,
$A\times B =\{(a,b):\forall (a\in A , b\in B) \}$.
If this symbolic definition is translated into words, it reads:"The cartesian product of two sets $A$ and $B$ is defined as the set of all ordered pairs $(a,b)$ such that for all $a$ belongs to $A$ and $b$ belongs to to $B$. 
My concern is about the last part in the symbolic definition, that is, "$:\forall (a\in A , b\in B)$". If this is translated into words it reads "such that for all $a$ belongs to $A$ and all $b$ belongs to to $B$". This sounds incomplete to me.
Is this correct both mathematically and grammatically? Would not it be better to define the cartesian product of $A$ and $B$ as $A\times B =\{(x,y):(x,y)=(a,b)\forall (a\in A , b\in B) \}$ or simply as $A\times B =\{(a,b):a\in A , b\in B) \}$. Among these three which is correct grammatically as well as rigorous mathematically?

REPLY [2 votes]: Formally speaking (since you ask for a "set theoretic definition") the cartesian product cannot be defined: it needs its own axiom in set theory. 
You cannot define sets with general formulas, otherwise you run the risk of running into Russell's paradox:
$$
R = \{x\colon x \not \in x\}.
$$
To avoid these kinds of open definitions one can only define subsets with a formula, as in:
$$
A = \{x \in B\colon \phi(x)\}.
$$
So the proposed definition (which is correct in naive set theory)
$$
A \times B := \{(a,b) \colon a \in A, b \in B\}
$$
is not actually a valid definition.
The solution is to say that $A \times B$ exists by assumption (precisely the axiom of pairing in ZF and most importantly, as pointed out in the comments, the axiom of power set) and satisfies the following relation (updated):
$$
x \in A \times B \iff \exists a \in A, \exists b\in B\colon x = (a,b).
$$
(added) I write here the nice formal definition given by @daniel-schepler in the comments: assuming you define $(a,b):=\{\{a\},\{a,b\}\}$, then actually you would say 
$$
A \times B := \{ p \in \mathcal P(\mathcal P(A \cup B))\colon \exists a\in A, \exists b\in B\colon p=(a,b)\}.
$$