I am working through a question on computing cardinals, and I came across this. I want to check that this is the correct way to solve this.
$\aleph_0^1=\aleph_0$, where $\aleph_0 = |\mathbb{N}|$
I can say that $\aleph_0^1 = |\mathbb{N}^1|$
Using this I can say there exists function that maps 1 to all the natural numbers, $\phi: 1 \rightarrow \mathbb{N}$
This function is injective, since for $a,b\in {1}$, $f(a)=f(b)\;\implies \; a=b \;$ (clearly true since a=b is always true)
This function is surjective, since $\forall \; n\in \mathbb{N}, \exists\; 1 \in 1 \implies f(1)=n \; $ (Clearly true, since 1 maps to all $\mathbb{N}$
The function is bijective, as it is a surjective and injective function, and thus $ |\mathbb{N}^1| \sim |\mathbb{N}|$
Therefore $\aleph_0^1 = \aleph_0$