Is it possible to create division via Set Theory?

Question

I've been reading a book on Set Theory (Charles C. Pinter), and it says,

...set theory is recognized to be the cornerstone of the "new" mathematics... [emph. added]

and that

...we can still form all the sets essential for mathematics... [emph. added]

Background

The book also states that Set Theory can be used to form the framework for mathematics (I am regrettably unable to quote Pinter on that). For example, if $A$ and $B$ are two disjoint sets, and that $a=\# A$ and $b=\# B$ denote that $a$ and $b$ are the cardinal numbers representing $A$ and $B$ respectively, then: $$a+b=\#(A\cup B),$$

and that $$a\cdot b=\#(A\times B).$$

Exponentiation can be produced by:

Let $a$ and $b$ be cardinal numbers, and let $I$ be a set such that $b=\#I$. If $a=a_i$,$\forall i\in I$, then:

$$a^b=\bigotimes_{i\in I}a_i.$$

where

$$\bigotimes_{i\in I}a_i=\#\left(\prod_{i\in I}A_i\right),$$

when $A_i$ is a family of sets, $a_i=\# A_i, \forall i\in I$.

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Question

This, oddly enough, is where Pinter decided to stop. He did not say how to produce division, or even the subtraction of cardinal numbers. I guess the subtraction of a cardinal would be (assuming that $0=\varnothing$, $1=\{\varnothing\}$, $2=\{\varnothing,\{\varnothing\}\}$... or that $0=\varnothing$, $1=\{0\}$, $2=\{0,1\}$...), and if $a=\#A$ and $b=\#B$: $$a-b=A\setminus B,$$ but this, to my knowledge, only works when $a\geq b$, for one cannot have a cardinal (let's say $x$) where $x<0$ (because that means the set is a proper subset of $\varnothing$, which is impossible).

How can I use Set Theory to represent functions like subtraction and division?

Answer 1

But how do you define subtraction or division anyway?

$a=c-b$ if and only if $a+b=c$

The same

$a=c/b$ if and only if $a \cdot b=c$

That is the most general definition possible.

Subtraction and division might not have a solution as they do not, for example, if you restrict to positive integers and integers respectively.

So you can define subtraction and division, yet the fact that you cannot perform that operation for all possible sets out there is equally valid.