I am having trouble grasping ideas for the following three problems (and I am unsure whether the third condition even holds). For arbitrary ordinals $\alpha$, $\beta$,
(Edit: on the LHS the operations are ordinal arithmetic, and the order relation between ordinals is given using Cartesian product of two sets, and giving it anti-lexicographical order)
- $\operatorname {card}(\alpha+\beta) = \operatorname {card}(\alpha)+\operatorname {card}(\beta)$
- $\operatorname {card}(\alpha\beta) = (\operatorname {card}(\alpha))(\operatorname {card}(\beta))$
- $\operatorname {card}(\alpha^\beta) = \operatorname {card}(\alpha)^{\operatorname {card}(\beta)}$
Cardinal number is defined as the minimal ordinal that is equipotent to a set $X$ and equality condition for cardinals is as follows: $$\beta \leq \alpha, \beta \approx \alpha \to \beta = \alpha$$
Do I need to find an explicit bijection between two sets having the speicified cardinalities? I feel like I am missing out on some important parts. Thank you in advance.