Exercises for Week 5

1. Following Kunen, we define $A \preceq B$ to mean that there is an injection from $A$ to $B$, and we define $A \approx B$ to mean that there is a bijection between $A$ and $B$. Prove that, for every ordinal $\beta$ and every NONEMPTY set $A$, the following statements are equivalent:
   1. $A \preceq \beta$;
   2. $A \approx \alpha$ for some ordinal $\alpha \le \beta$;
   3. There is a surjection from $\beta$ onto $A$.
2. Recall that $\omega_0 = \omega$, $\omega_{\alpha+1} = (\omega_\alpha)^+$, and $\omega_\lambda = \sup_{\alpha < \lambda} \omega_\alpha$ if $\lambda$ is a limit ordinal. In class we sketched the proof that $\omega_{\alpha+1}$ is a cardinal. Prove that $\omega_\lambda$ is a cardinal if $\lambda$ is a limit ordinal.
3. For sets $A$ and $B$, recall that $A^B$ denotes the set of all functions from $B$ to $A$. Prove ONE of the following statements. (Both are true, but you are asked to prove only one as homework.)
   1. $A^{B \cup C} \approx A^B \times A^C$ if $A$ and $B$ are disjoint.
   2. $(A^B)^C \approx A^{B \times C}$.
4. Prove that a countable union of countable sets is countable. That is, if $S \preceq \omega$ and $\forall X \in S\,(X \preceq \omega)$, then $\bigcup S \preceq \omega$. You may use the Axiom of Choice (or the Well-ordering Theorem, or Zorn's Lemma.)