# Prove that the set with $c^a$ is countable

$A = \{$prime integers$\}$
$B = \{$odd, positive multiples of $5$$\}$
$C = \{(b^a)/(a^b)|a \in A, b \in B\}$

Prove that $C\cup \{c^a|a \in A, c \in C\}$ is countable.

I have no idea how to go about this. Can I say that since a subset of a countable set is countable, $C$ is therefore countable: it maps onto $\mathbb{N}$ and so I am essentially just proving $\mathbb{N}^a$? This is correct? How would I prove even that part?

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It looks like your set is a subset of the rational numbers. – Braindead Jan 26 '14 at 4:27
@Braindead I am suffering from a condition similar to your username, and cannot figure out how to apply your comment to the proof. Is it enough to just say that if it's a subset of a countable set, it's therefore also countable? Do I actually need to care about what's inside set A and B in this question? – idlackage Jan 27 '14 at 16:02
It is true that a subset of a countable set is countable. Is this a homework problem for a class, or are you just doing this problem for your own personal understanding? – Braindead Jan 27 '14 at 17:42
A more challenging problem would be to have $a^c$ rather than $c^a$. – Braindead Jan 27 '14 at 17:42
@Braindead It is a homework question, but it would also be very helpful for personal understanding. Why is $a^c$ more challenging than $c^a$? – idlackage Jan 28 '14 at 0:07