Which of the following statements are true, and which false? Justify your answers.

(a) For any ordinals $\alpha$ and $\beta$ with $\beta \neq 0$, there exist ordinals $\gamma$ and $\delta$ with $\delta<\beta$ such that $\alpha=\beta . \gamma+\delta$.

(b) For any ordinals $\alpha$ and $\beta$ with $\beta \neq 0$, there exist ordinals $\gamma$ and $\delta$ with $\delta<\beta$ such that $\alpha=\gamma \cdot \beta+\delta$.

(c) $\alpha \cdot(\beta+\gamma)=\alpha \cdot \beta+\alpha \cdot \gamma$ for all $\alpha, \beta, \gamma$.

(d) $(\alpha+\beta) \cdot \gamma=\alpha \cdot \gamma+\beta \cdot \gamma$ for all $\alpha, \beta, \gamma$.

(e) Any ordinal of the form $\omega . \alpha$ is a limit ordinal.

(f) Any limit ordinal is of the form $\omega . \alpha$.