On Halmos' "Naive Set Theory" he states that there's a natural one-to-one correpondence between a set $X \times Y$ and a certain set of families.
"Consider, indeed, 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 the promised one-to-one correspondence.
Trying to come to grips with this, I used two families $z'$ and $z''$, with $Z=\{z', z''\}$.
In this example we could have
$z'=\{(a, G_{a}), (b, G_{b})\} \ \ \ \ \ \ \ \ \text{and} \ \ \ \ \ \ \ \ z''=\{(a, H_{a}),(b, H_{b})\};$
and then, $z'_{a}=G_{a}\ ,\ \ \ z'_{b}=G_{b}\ ,\ \ \ z''_a=H_a\ ,\ \ \ z''_b=H_b.$
I presumed in this case $X=\{G_a, H_a \}$ $Y=\{G_b, H_b \}$ $X\times Y=\{(G_a, G_b), (G_a, H_b), (H_a, G_b), (H_a, H_b)\}.$
But, $f(z')=(G_a, G_b)$ and $f(z'')=(H_a, H_b)$ and thus, $f$ wouldn't be a one-to-one correspondence. Where's the mistake? Thanks.
If it'd be of any help, there's a link to another question on this section.