Regarding whether one has a circular definition in the following case

Question

In "Principles of Analysis" by Charalambos Aliprantis on pg. 6, he defines the Cartesian product of a family of sets as the set of all functions from the index set to the union of that family satisfying a certain condition. If we define a function to be a certain subset of the Cartesian product of two sets and then use the function to define the Cartesian product of a family of sets indexed by some indexed set, aren't we committing the error of making a circular definition? I may be wrong, in fact I think I am since may be we can distinguish between the Cartesian product of two sets and the Cartesian product of a family of sets. But I just want some confirmation from an expert. Many thanks in advance

Answer 1

I don't know that specific book that you mention, but your objection is correct: you can't define Cartisian product using the concept of function and als to define function using the concept of Cartesian product.

However, note the the concept of function only uses the concept of Cartesian product of two sets $A$ and $B$. And you can define it as$$\left\{\,\{a,\{a,b\}\}\middle|\,a\in A\wedge b\in B\right\}.$$