[Throughout this question, assume the axiom of choice.]

Let $\kappa, \lambda$ and $\mu$ be cardinals. Define $\kappa+\lambda, \kappa \lambda$ and $\kappa^{\lambda}$. What does it mean to say $\kappa \leqslant \lambda$ ? Show that $\left(\kappa^{\lambda}\right)^{\mu}=\kappa^{\lambda \mu}$. Show also that $2^{\kappa}>\kappa$.

Assume now that $\kappa$ and $\lambda$ are infinite. Show that $\kappa \kappa=\kappa$. Deduce that $\kappa+\lambda=\kappa \lambda=\max \{\kappa, \lambda\}$. Which of the following are always true and which can be false? Give proofs or counterexamples as appropriate. (i) $\kappa^{\lambda}=2^{\lambda}$; (ii) $\kappa \leqslant \lambda \Longrightarrow \kappa^{\lambda}=2^{\lambda}$; (iii) $\kappa^{\lambda}=\lambda^{\kappa}$.