hi, just wanted to ask: is there a logical-ordered listing of the subjects in planetmath? The ordering modes that I've found in the site was order by subject index or alphabetically.
i'm looking for a list that enumerates all the prerequisite concepts before introducing a new one like:
01-mappings
02, 03-surjective map & injective map
04-bijective map
05-set of natural numbers
06-countable set
Hi there,
I am looking for a faster algorithm for counting the number of data points that are close to a particular point in k-dimensional hyperspace? The current method of calculating all the distances to data points is too slow for my application to handle.
(If this should be posted elsewhere then please let me know)
Thanks in advance
For #Y = 0 the cartesian product is the empty set and the proposition is trivially true.
We try to prove #(X2×Y2) = #X2 ⋅ #Y2 for #Y2 = #Y1++
Let's choose an arbitrary element a2 from Y2 then #(Y2 \ {a2}) = #Y1
X2×Y2 = X2×(Y2{a2}) ∪ X2×{a2}
as both sets are disjoint #(X2×Y2) = #(X2×(Y2{a2})) + #(X2 × {a2})
and #(X2×(Y2{a2})) = #(X1×Y1{a2})
j: X2 → X2 ×{a2} = j(x) = (x, a2) for x in X2 is a bijection
so #(X2×{a2}) = #X2
and #(X2×Y2) = #(X2×(Y2{a2})) + #(X2×{a2}) = #(X1×Y1) + #X1 =
= #X1⋅#Y1 + #X1 = #X1⋅#Y1++ = #X2⋅#Y2
Hello! It's my first time here. Please be kind!
First off, thank you for a monumental effort. It's almost like Bourbaki reincarnated for the masses!
Is there a Makefile on the planetmath repo to create the true local copy of planetmath with cross-linking sections, indexed items etc.?