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Jan Meyer
2026-04-28 13:50:38 +02:00
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10 Courses/01 - WiSe 2025_26/DAS/Formulas.md
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@@ -1,6 +1,6 @@
## Asymptotic Equivalence Classes (Big-O) ## Asymptotic Equivalence Classes (Big-O)
The equivalence relation definition given in the task is asking you to find functions that grow at the exact same rate (also known as Big-Theta $\Theta$): The equivalence relation definition given in the task is asking you to find functions that grow at the exact same rate (also known as Big-Theta $Theta$):
$f asymp g <==> f in O(g) and g in O(f)$ $f asymp g <==> f in O(g) and g in O(f)$
@@ -44,7 +44,7 @@ $$
&15^5 = 759,375 && "--- in the range / below" 10^6 \ &15^5 = 759,375 && "--- in the range / below" 10^6 \
&16^5 = 1,048,576 && "--- outside the range / above" 10^6 \ &16^5 = 1,048,576 && "--- outside the range / above" 10^6 \
&=> 15 "numbers in the form "x^5"exist" && &=> 15 "numbers in the form "x^5"exist" &&
$$
> [!warning] > [!warning]
> Now, we can't add $1,000$ and $15$, since there are numbers that match both, so we need to subtract these duplicates. > Now, we can't add $1,000$ and $15$, since there are numbers that match both, so we need to subtract these duplicates.
@@ -54,6 +54,7 @@ $$
& 4^10 = 1,048,576 \ & 4^10 = 1,048,576 \
& => 3 "numbers that are both" x^2 "and" x^5 "exist" & => 3 "numbers that are both" x^2 "and" x^5 "exist"
$$ $$
#### Final calculation: #### Final calculation:
Formula: $"Elements that are" x^2 + "Elements that are" x^5 - "Elements that are both"$ Formula: $"Elements that are" x^2 + "Elements that are" x^5 - "Elements that are both"$
$==> 1,000 + 15 - 3 = 1,012$ $==> 1,000 + 15 - 3 = 1,012$