Cardinality of a set of functions from $\mathbb N$ to a set of real numbers

Question

Let $S$ be the set of functions from $\mathbb N$ to the set $\{\sqrt{2}, \sqrt{3}, \sqrt{5}, \sqrt{7}\}$. Determine $|S|$.

I understand how to prove the converse case where $T$ is the set of functions from $\{\sqrt{2}, \sqrt{3}, \sqrt{5}, \sqrt{7}\}$ to $\mathbb N$ by defining a bijective function $f:T \to \mathbb N \times \mathbb N \times \mathbb N \times \mathbb N$, where $|\mathbb N \times \mathbb N \times \mathbb N \times \mathbb N| = |\mathbb N|$. How would I approach this case though?

score 3 · Accepted Answer · edited Dec 08 '14 at 22:22

3

Do you know how to calculate the cardinality of all $0$-$1$ sequences?

edited Dec 08 '14 at 22:22

Michael Hardy

1

answered Dec 08 '14 at 21:49

Jytug

3,169

Would that mean defining a bijective function $f:$ $S$ $\to$ $[0, 1]$ x $[0, 1]$ x $[0, 1]$ x $[0, 1]$? – user198693 Dec 08 '14 at 21:56
Well, by $ 0 -1 $ sequences I mean functions from $ \mathbb{N}$ to ${0,1}$. It's easier to count them, but once you have done that, it should be easier to do your task – Jytug Dec 08 '14 at 22:00
$ \textbf{Hint:}$ every real number has its binary representation – Jytug Dec 08 '14 at 22:03
I think I understand it now. |S| = |P($\mathbb N$)| = |$\mathbb R$|? – user198693 Dec 08 '14 at 22:05
That's correct. – Jytug Dec 08 '14 at 22:41

Cardinality of a set of functions from $\mathbb N$ to a set of real numbers

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