I'm having a lot of problem defining my sets for a inclusion/exclusion problem:
How many positive integers below 10 000 are not perfect square and are not a power of 3,4,5... of an integer
So first I let $$P_i = \{n\in \mathbb{Z} | 1\leq n \leq 9999, n = k^i, k\in \mathbb {Z}\}$$ for $i\geq 2$
But from here I get a bit confused.
We use the notation $N(P_i)$ for the number of elements in the set $ P_i$ . But I figured it would mean that $N(P_i) = \lfloor\sqrt[i]{9999}\rfloor$ where the greatest $i$ would be $13$. Would I be wrong? I used the fact that $\lfloor\sqrt[i]{9999}\rfloor \geq 2$.
Moreover, after I sum all the floor function for $1\leq i \leq 13$ how do I define the intersections? For example, if an integer is a power of $2$ and $3$ how could I define the set of all the positive integers below $9999$ that have both these properties?
And finally, what would be the underlying logic for $N(P_i) = \lfloor\sqrt[i]{9999}\rfloor$. Is it that the greatest integer such that it is the $i^{th}$ power of an integer $k$ would be that one, cause all subsequent one when elevated to the $j \geq i+1$ power would be over 9999...
I think my question about my confusion is confusing but if someone could help me to define my sets and find their cardinality I would be so thankful.
Thank you