Showing that a set is well-defined

[Mr Davis 97]

Homework Statement

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

Homework Equations

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?

 

[fresh_42]

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 learned 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.