vault backup: 2026-03-09 18:38:49

DAS/Counting.md

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+## 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: $"Elements that are" x^2 + "Elements that are" x^5 - "Elements that are both"$
+$==> 1,000 + 15 - 3 = 1,012$
+
+### For two sets
+$|"Total possibilities"| - |"Avoid 1"| - |"Avoid 2"| + |"Avoid Both"|$
+Where 
+- $"Avoid 1"$ are all elements *not matching* condition 1
+- $"Avoid 2"$ are all elements *not matching* condition 2
+- $"Avoid Both"$ are all elements *not matching* condition 1 **and** 2
+
+## Type of Task: Boolean Lattice
+Find the amount of upper bounds in ${0, 1}^5$ for the set of vectors $S = {x, y, z}$
+$$
+& x = (0, 1, 0, 0, 0) \
+& y = (0, 0, 1, 0, 1) \ 
+& z = (0, 1, 1, 0, 0) \
+$$
+> [!INFO]
+> Method used is called **All-Zero Column Method**
+
+We look for columns that have $0$'s in all three vectors: Column $1$ and $4$, that's $2$ columns, so we define $k = 2$.
+To get our result we calculate $2^k$ since there are only two possible states for each value $(0, 1)$: 
+$2^2 = 4$.
+So we have **4** upper bounds in total.
+
+> [!INFO]
+> For the amount of lower bounds we'd check for all-ones columns