Shrinking a family of sets preserving cardinality

Question

I'm having trouble with a simple problem about infinitary combinatorics. Suppose $\kappa$ is an infinite cardinal, $\mathcal A=\{A_\alpha: \alpha<\kappa\}$ and for each $\alpha$, $|A_\alpha|=\kappa$. I'm trying to show that there exists a family $(A_\alpha': \alpha<\kappa)$ such that for each $\alpha$, $A'_\alpha\subset A_\alpha$, $|A_\alpha'|=\kappa$ and such that for every $\alpha, \beta<\kappa$, if $\alpha\neq \beta$ then $A_\alpha\cap A_\beta=\emptyset$. I think it's possible, but I'm not sure. I tried to proceed by transfinite induction but I'm having trouble.

Answer 1

I think I got it. Let $f:\kappa\rightarrow \kappa\times \kappa$ be one-to-one and onto, and let $f=(g, h)$.

We construct the sets $A_\alpha'=\{a_\alpha^\eta:\eta<\kappa\}$ proceeding by induction on $\xi$, as follows: If $a_{g(\gamma)}^{h(\gamma)}$ has been defined for every $\gamma<\xi$ for some $\xi<\kappa$, let $a_{g(\xi)}^{h(\xi)}\in A_{g(\xi)}\setminus\{a_{g(\gamma)}^{h(\gamma)}: \gamma<\xi\}$.