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?
Best Answer
The issue is of that of free variables. When you write $\{x : \varphi\}$, it is expected that $x$ is the only free variable of $\varphi$ (there might be parameters fixed in advance).
Once you put a quantifier on $a$ and $b$, they are no longer free, and the formula is now a sentence which is either "always true" or "always false". So the set is either everything or empty.
To add insult to injury, $\forall(a\in A, b\in B)$ is not a well-formed formula to begin with. The comma, while not a valid logical symbol is often taken to mean "and" in the set-builder context.
So indeed the correct way to write this would be, as others and yourself have suggested $$A\times B=\{(a,b): a\in A, b\in B\}.$$