27 lines
954 B
Markdown
27 lines
954 B
Markdown
## Inclusion-Exclusion Principle
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Principle that dictates that when combining / overlapping sets, you have to make sure to not include elements that occur in multiple sets multiple times.
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### Example:
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How many integers between $1$ and $10^6$ are of the form $x^2$ or $x^5$ for some $x in NN$?
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#### How many $x^2$?
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$sqrt(10^6) = 10^3 = 1.000$
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#### How many $x^5$?
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By estimation:
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$$
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&15^5 = 759,375 && "--- in the range / below" 10^6 \
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&16^5 = 1,048,576 && "--- outside the range / above" 10^6 \
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&=> 15 "numbers in the form "x^5"exist" &&
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$$
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> [!warning]
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> Now, we can't add $1,000$ and $15$, since there are numbers that match both, so we need to subtract these duplicates.
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#### How many $x^2$ and $x^5$ / $x^10$?
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$$
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& 3^10 = 59,049 \
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& 4^10 = 1,048,576 \
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& => 3 "numbers that are both" x^2 "and" x^5 "exist"
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$$
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#### Final calculation:
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Formula: $"Elements that are" x^2 + "Elements that are" x^5 - "Elements that are both"$
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$==> 1,000 + 15 - 3 = 1,012$
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