# Generalization of the concept of Cartesian product in Halmos' book.

On page 36 of his Set Theory book, Halmos states the following (with $$X$$ and $$Y$$ being two sets whose cartesian product we'll consider):

"Consider any particular unordered pair {a, b} with $$a\neq b$$, and consider the set Z of all families z, indexed by {a, b}, such that $$z_a \in X$$ and $$z_b \in Y$$. If the function $$f$$ from $$Z$$ to $$X \times Y$$ is defined by $$f(z) = (z_a, z_b)$$, then $$f$$ is a one-to-one correspondence between $$X \times Y$$ and $$Z$$."

My question is: where is this one-to-one correspondence? The way I'm interpreting this is the following:

We have a set of families $$Z$$, which is the range of some family {$$z_i$$}, and where each term of this family ($$Z$$) is also a family. Then, the domain of such family {$$z_i$$} is the index $$I=\,$${$$a, b$$}; such that the terms of the family will be $$z_a$$ and $$z_b$$. Then, if $$z_a \in X$$ and $$z_b \in Y$$, if we define a function $$f: Z\longrightarrow X\times Y$$, where $$f(z):= (z_a, z_b)$$, then, whatever $$z$$ we choose from $$Z$$, will its image be the fixed ordered pair $$(z_a, z_b)$$?

Sorry if I'm overcomplicating this, I just started studying families and this has me a bit confused.

A family of elements indexed by the two element set $$\{a,b\}$$ is just a function with domain $$\{a,b\}$$, and for such a function $$z$$, the index notation simply means $$z_a=z(a),\ z_b=z(b)$$. So the set $$Z$$ is $$Z=\{z:\{a,b\}\to X\cup Y: z(a)\in X,\,z(b)\in Y\}\,.$$ The correspondence to $$X\times Y$$ is given by $$z\mapsto (z(a),\,z(b))$$ and $$(x,y)\mapsto (a\mapsto x,\ b\mapsto y)$$.
• The first map goes $Z\to X\times Y$, the other one is its inverse $X\times Y\to Z$, and $\mapsto$ denotes assignment. – Berci Dec 31 '20 at 23:15