## Inclusion-Exclusion Principle
 Principle that dictates that when combining / overlapping sets, you have to make sure to not include elements that occur in multiple sets multiple times.

### Example:
How many integers between $1$ and $10^6$ are of the form $x^2$ or $x^5$ for some $x in NN$?
#### How many $x^2$?
$sqrt(10^6) = 10^3 = 1.000$
#### How many $x^5$?
By estimation:
$$
&15^5 = 759,375 && "--- in the range / below" 10^6 \
&16^5 = 1,048,576 && "--- outside the range / above" 10^6 \
&=> 15 "numbers in the form "x^5"exist" &&
$$
> [!warning]
> Now, we can't add $1,000$ and $15$, since there are numbers that match both, so we need to subtract these duplicates.

#### How many $x^2$ and $x^5$ / $x^10$?
$$
& 3^10 = 59,049 \
& 4^10 = 1,048,576 \
& => 3 "numbers that are both" x^2 "and" x^5 "exist"
$$
#### Final calculation:
Formula: $"Elements that are" x^2 + "Elements that are" x^5 - "Elements that are both"$
$==> 1,000 + 15 - 3 = 1,012$