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.?
I discovered algebraic general topology and a generalization of limit for arbitrary functions: https://mathematics21.org - congratulate me
congratulations
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
and it works&
:point_up: Edit: Hello there. I have had a very long hiatus from PlanetMath. This is mainly because I left college and was too busy with my job too keep up with PlanetMath anymore. Due to circumstances that I would prefer not to get into, I now have much more time, and I decided to check out PlametMath. To my consternation, I saw that there were many unfavorable changes. These include:
- The list of entries is difficult to read.
- The method of forcing links has either changed or been eliminated. Many of my forced link commands come up as unlinked URL's in my entries.
- I do not know how to edit my entries, or if that is even possible enymore.
- I do not know how to access the LaTeX code that someone used to create an entry.
Out of all of these, number 4 is my biggest concern, as there are some LaTeX techniques, primarily for compass and straightedge constructions as well as entries dealing with the Beltrami-Klein model of hyperbolic geometry, that I have forgotten. If anyone could tell me how I can access that LaTeX code, I would greatly appreciate it.
Assuming that issues 2 and 3 get fixed, I would love to go through my entries and make them look more visually appealing. I do not plan on adding any new content, though, as long as issue 1 is present. I realize that the appearance of the PlanetMath site cannot cater to me; however, I simply cannot bring myself to add more clutter to a format that is already clunky and visually unappealing.
Any input that anyone has, especially someone who has been active with PlanetMath and has witnessed the many changes that have occurred.
:point_up: Edit: Hello there. I have had a very long hiatus from PlanetMath. This is mainly because I left college and was too busy with my job too keep up with PlanetMath anymore. Due to circumstances that I would prefer not to get into, I now have much more time, and I decided to check out PlametMath. To my consternation, I saw that there were many unfavorable changes. These include:
- The list of entries is extraordinarily difficult to read.
- I do not know how to access a list of entries consisting solely of entries in which I have/had editing powers.
- The method of forcing links has either changed or been eliminated. Many of my forced link commands come up as unlinked URL's in my entries.
- I do not know how to edit my entries, or if that is even possible anymore.
- I do not know how to access the LaTeX code that someone used to create an entry.
Out of all of these, number 5 is my biggest concern, as there are some LaTeX techniques, primarily for compass and straightedge constructions as well as entries dealing with the Beltrami-Klein model of hyperbolic geometry, that I have forgotten. If anyone could tell me how I can access that LaTeX code, I would greatly appreciate it.
Assuming that issues 3 and 4 get fixed, I would love to go through my entries and make them look more visually appealing. This process would be even smoother for me if issue 2 were resolved. I do not plan on adding any new content, though, as long as issue 1 is present. I realize that the appearance of the PlanetMath site cannot cater to me; however, I simply cannot bring myself to add more content to a format that is already clunky, cluttered, and visually unappealing.
I would appreciate any input that anyone has, especially from someone who has been active with PlanetMath and has witnessed the many changes that have occurred. Again, I would especially appreciate it if someone knew how to view the LaTeX code used to create an entry and would share that information with me. I look forward to hearing from you.
:point_up: Edit: Hello there. I have had a very long hiatus from PlanetMath. This is mainly because I left college and was too busy with my job too keep up with PlanetMath anymore. Due to circumstances that I would prefer not to get into, I now have much more time, and I decided to check out PlanetMath. To my consternation, I saw that there were many unfavorable changes. These include:
- The list of entries is extraordinarily difficult to read.
- I do not know how to access a list of entries consisting solely of entries in which I have/had editing powers.
- The method of forcing links has either changed or been eliminated. Many of my forced link commands come up as unlinked URL's in my entries.
- I do not know how to edit my entries, or if that is even possible anymore.
- I do not know how to access the LaTeX code that someone used to create an entry.
Out of all of these, number 5 is my biggest concern, as there are some LaTeX techniques, primarily for compass and straightedge constructions as well as entries dealing with the Beltrami-Klein model of hyperbolic geometry, that I have forgotten. If anyone could tell me how I can access that LaTeX code, I would greatly appreciate it.
Assuming that issues 3 and 4 get fixed, I would love to go through my entries and make them look more visually appealing. This process would be even smoother for me if issue 2 were resolved. I do not plan on adding any new content, though, as long as issue 1 is present. I realize that the appearance of the PlanetMath site cannot cater to me; however, I simply cannot bring myself to add more content to a format that is already clunky, cluttered, and visually unappealing.
I would appreciate any input that anyone has, especially from someone who has been active with PlanetMath and has witnessed the many changes that have occurred. Again, I would especially appreciate it if someone knew how to view the LaTeX code used to create an entry and would share that information with me. I look forward to hearing from you.
