# Cartesian Product-Need some help on a proposed construction

I'm trying to construct a rigorous set theoretic formulation of the old formulation of functions in terms of variables. I began with the following old school definition of variable:

Definition XI: A variable over a set S is an unspecified individual element of a set. The set S is called the range of the variable, any specified element of the set S is called a value of the variable.

This leads to the following definition of function given by Arnold Dresden:

Definition XVI: When the two variables x and y are so related that to each value of the range of x there corresponds a value of y, we say that y is a function of x. Notation: y= f (x).

This is more or less the old Dirichet definition of function given in the slightly more precise language of variables.

Here's my proposed "rigorous" definition of variables:

Def: Let x be a symbol. Let S be a nonempty set where x can be assigned any member of S as it's value. Then we define the variable U with range S as follows: U = {x} X S = { (x,a) : $$a \in S$$} Note U is a relation in the set theoretic sense since every ordered pair has the same first member, namely x.

I'm assuming the Kuratowski definition of an ordered pair i.e. (x,a) = { {x}, {x,a}}

Now here's where I need someone to double check me. Let U be the variable whose range is the domain S of a function $$f:S\rightarrow T$$ and let V be the variable whose range is the functions's range. U = {x} X S = { (x,a) : $$a \in S$$} V = {y} X T = { (y,b) : $$a \in T$$}

I want to now define a function between variables x on the domain and the range y using these definitions of variables.

Here's my question: How do I calculate U X V?

Here's my computation, which I'm pretty sure is wrong:

U X V = ( {x} X S) X ({y} X T) = (x,y) X (S X T).

Something looks VERY wrong here. Please someone take another look.

{x}×S × {y}×T = { ((x,s), (y,t)) : s in S, t in T }

• That was my initial computation, but I was a bit confused. We simply end up with a set of ordered pairs that's the union of the variables U and V? If so, then the definition of f is very simple: It's the subset of U X V where no 2 different ordered pairs have the same first member i.e. (($x_1$, $s_1$), (y,t)) = (($x_2$, $s_2$), (y,t)) iff $x_1$ =$x_2$ AND $s_1$ = $s_2$. Which is relatively simple if one wants to continue using the vocabulary of variables.in modern set theoretic language, but not as simple as working directly with set constructions. Nov 18, 2019 at 0:07
• if this is correct,then the problem was I'm overthinking this. We just build each ordered pair in U X V by taking an ordered pair of U as the first member and the "corresponding" ordered pair of V as the second member. Nov 18, 2019 at 0:13