I am trying to prove that the set of all functions from the set of even numbers into $\{a, b, c \}$ is uncountable.
I know I need to treat functions as series and start from there somehow (similarly to proving that set of $f: N \to \{0,1\}$ is uncountable) but I am having a problem with applying Cantor's diagonal argument in this particular case.
Can you please give me any hints?
Answer
What you should realize is that each such function is also a sequence.
The diagonal arguments works as you assume an enumeration of elements and thereby create an element from the diagonal, different in every position and conclude that that element hasn't been in the enumeration.
To be concrete assume that $f_n$ is such an enumeration and consider the function
$$\phi(2n) = \begin{cases}b & \mbox{if } f_n(2n) = a\\ a & \mbox{otherwise}\end{cases}$$
Since $\phi(2n) \ne f_n(2n)$ we have that $\phi\ne f_n$.
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