- #1

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## 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?