vault backup: 2026-03-09 18:27:32

Counting.md

@@ -24,3 +24,21 @@ $$
 #### Final calculation:
 Formula: $"Elements that are" x^2 + "Elements that are" x^5 - "Elements that are both"$
 $==> 1,000 + 15 - 3 = 1,012$
+
+## Type of Task: Boolean Lattice
+Find the amount of upper bounds in ${0, 1}^5$ for the set of vectors $S = {x, y, z}$
+$$
+& 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