# Showing that a set is well-defined

## Homework Statement

Assume ##A## and ##B## are given sets, and show that there exists a set ##C## such that ##C = \{\{x\} \times B : x \in A\}##.

## The Attempt at a Solution

To show that this set exists, I thought to maybe use the subset axioms (or the axiom schema of restricted comprehension). I claim that if ##C^* = \{w \in \mathcal{P} (A \times B) : w = \{x\} \times B ~\text{for some}~ x \in B \}##, then ##C^*=C##. First ##C^*## is well-defined by the subset schema. Additionally, it contains all elements of the desired sort because if we fix ##x \in A##, then clearly ##\{x\} \times B \subseteq A \times B##, so ##\{x\} \times B \in \mathcal{P} (A \times B)##. So ##C^* = C##, and hence ##C## is a well-defined set.

Is this a correct argument? Is there any way I could make it flow better?

## Answers and Replies

fresh_42
Mentor
It would help a lot, if you
1. said what exactly has to be shown, since the existence appears to be already given by the fact, that you wrote down the set in a constructive way.
2. said what the axioms are, e.g. via a link or a reference, especially because I doubt they are everywhere the same ones or that anybody learnt them by heart
3. said, what **** ##\mathcal{P}(A)## is. I suppose it is meant to be the power set of ##A##, but this is not the only way to denote a power set, so a definition would be at least polite
4. checked, whether there is a typo in your definition of ##C^*##. I assume there is.
At least those where the points which prevented me from having a closer look.