If $X$ and $Y$ are nonempty sets, we define $Y^X = \{f\,|\,f\colon X\to Y\}$ and $\mathrm{card}(Y^X)=\mathrm{card} (Y)^{\mathrm{card}(X)}$.
- Show that the above definition is independent on the sets representing cardinal numbers, that is, if $X'$ and $Y'$ are sets with the same cardinality as $X$ and $Y$, respectively, then $\mathrm{card}(Y)^{\mathrm{card}(X)} = \mathrm{card}(Y')^{\mathrm{card}(X')}$.
- Prove that $1^c = 1$ and $c^1 = c$, where $c$ is a cardinal number.
For 1, saying $X'$ has the same cardinality as $X$ means there is a bijection between them. You are expected to find a bijection between $Y^X$ and $Y'^{X'}$ given that there is one between each of the basic pairs.
For 2, you are expected to find an explicit bijection between the set of functions on the left and the cardinal number on the right.