Quantitative Cardinality Sets Project P&L: 0 (≃ 0 JPY)
Brining sets with quantifiabiable cardinality into common use in mathematics.
YAML
企画
This project shall be a variant of an initiative to define and introduce sets with negative (and more generally quantified) cardinality into mathematics and computing, based on a specific proposal, that I received by e-mail from a as-of-yet undisclosed thinker (let me know if I should disclose it publicly). It follows below:
{1,2}+{3,4}={1,2,3,4}
{1,2}+{2,3}={1,2,2,3}={1,2_2,3}
1,2+1,2=2*{1,2}={1,1,2,2}={1_2,2_2}
{a_x}+{a_y}={a_(x+y)}
{1,2,3}-{1}={2,3}
{1,2}-{1,2}={}={1_0,2_0}
{1,2}-{1,2,3,4}=-{3,4}={3_-1,4_-1}
{1,2,3}-{3,4,5}={1,2}-{4,5}={1,2,4_-1,5_-1}
{a_x}-{a_y}={a_(x-y)}
3*{1,2,3}={1,1,1,2,2,2,3,3,3}={1_3,2_3,3_3}
-2*{1,2,3}={1_-2,2_-2,3_-2}=-{1_2,2_2,3_2}
0.5*{1,2,3}={1_0.5,2_0.5,3_0.5}
2*{1_0.5,2_0.5,3_0.5}={1,2,3}
y*{a_x}={a_(x*y)}
{a,b}*{c,d}={a+c,a+d,b+c,b+d}
{a,b,c}*{d,e}={a+d,a+e,b+d,b+e,c+d,c+e}
{a_x,b_y}*{c_z,d_t}={(a+c)_xz,(a+d)_xt,(b+c)_yz,(b+d)_yt}
{{a},{b}}*{{c},{d}}={{a,c},{a,d},{b,c},{b,d}}
{{a},{b}}^2={{a_2},{a,b}_2,{b_2}}
P({a,b,c,d})，P({a,b}),P({c,d})：
P({a,b})={0,{a},{b},{a,b}}
P({c,d})={0,{c},{d},{c,d}}
P({a,b,c,d})={0,{c},{d},{c,d},{a},{a,c},{a,d},{a,c,d},{b},{b,c},{b,d},{b,c,d},{a,b},{a,b,c},{a,b,d},{a,b,c,d}}
P(A+B)=P(A)*P(B)
a_{b}=a+b。{a}*{b}={a+b}={a_{b}}
0={}
1={0}
2=1+1={0}+{0}={0,0}={0_2}
3=2+1={0,0}+{0}={0,0,0}={0_3}
n={0_n}。x={0_x}。
{2,4,6,...}/{1}={1,3,5,...}
{1,2,3,...,}/{2,4,6,...}={0,-1}
[0,∞)/[0,1)={0,1,2,3,...}
x_{a}=x+a,x_{b}=x+b,x_{c}=x+c。A={m,n,p}，{{a},{b},{c}}^A={{m+a},{m+b},{m+c}}*{{n+a},{n+b},{n+c}}*{{p+a},{p+b},{p+c}}。
{0,1}^A=P(A)
{0,0}^A=2^A=2^|A|
{1,1}^A={A_2^|A|}
{0,1,2}^{a,b,c}={0,{a},{a_2}}*{0,{b},{b_2}}*{0,{c},{c_2}}
{{c},{d}}^{a,b}={{a+c,b+c},{a+c,b+d},{a+d,b+c},{a+d,b+d}}
{{c,d},{e,f}}^{a,b}={{a+c,a+d},{a+e,a+f}}*{{b+c,b+d},{b+e,b+f}}
{{c},{d},{e},{f}}^{a,b}={{a+c},{a+d},{a+e},{a+f}}*{{b+c},{b+d},{b+e},{b+f}}
{a_x,b_y}+{a_z,b_t}={a_(x+z),b_(y+t)}
(a_x,b_y}*{c_z,d_t}={a+c_xz,a+d_xt,b+c_yz,b+d_yt}
{{c},{d}}^{a,b}={{a+c},{a+d}}*{{b+c},{b+d}}
P(A+B)=P(A)*P(B),A^(C*B)=(A^C)^B
P(A)={0,1}^{a_x,b_y,c_z}
P(A)={0,1}^{a_x}*{0,1}^{b_y}*{0,1}^{c_z}
{0,1}^{a}={0,{a}}，{0,1}^{a_x}=({0,1}^{a})^x，P(A)={0,{a}}^x*{0,{b}}^y*{0,{c}}^z
{0,{a}}^x={0,{a}_x,{a_2}_x*(x-1)/2,...,{a_n}_x*(x-1)*...*(x-n+1)/n!,.....}
P({a_x,b_y,c_z})={0,{a}_x,{a_2}_x*(x-1)/2,...,{a_n}_x*(x-1)*...*(x-n+1)/n!,.....}*{0,{b}_y,{b_2}_y*(y-1)/2,...,{b_n}_y*(y-1)*...*(y-n+1)/n!,.....}*{0,{c}_z,{c_2}_z*(z-1)/2,...,{c_n}_z*(z-1)*...*(z-n+1)/n!,.....}
1/{0,1}={0,1}^-1={0,1_-1,2,3_-1,4_1,.....}。
{0,1_-1,2,3_-1,4_1,.....}*{0,1}={0,1_-1,2,3_-1,4_1,.....,1,2_-1,3,4_-1,.......}={0}=1。1/{0,1,2}={0}+{1,2}*-1+{1,2}^2+{1,2}^3*-1+...，1-2+4-8+....=1/3。
1-n+n^2-n^3+....=1/(n+1)。
1/{0,1}^2={0}+{1}*-2+{2}*3+{3}*-4+....，1-2+3-4+....=1/4。
1/{0,1}^3={0}+{1}*-C(1,3)+{2}*C(2,4)+{3}*-C(3,5)+....，1-3+6-10+....=1/8。
C(0,n-1)-C(1,n)+C(2,n+1)-C(3,n+2)+.....=1/2^n。
C(n,n)*C(k,k+n-1)-C(n-1,n)*C(k+1,k+n)+,,,,+(-1)^i*C(n-i,n)*C(k+i,k+n-1+i)+......+(-1)^n*C(0,n)*C(k+n,k+2n-1)=0.
1/{-1,0_-1}=({-1}-1)^-1={1,2,3,4,.....}
1/{-2,0_-1}={2,4,6,8,.....}
1/{1,2,3,4,....}={-1,0_-1}
{1,2,3,....}/{2,4,6,....}={-2,0_-1}/{-1,0_-1}={0,-1}
{1,3,5,...}-{2,4,6,....}={1,2:-1,3,4:-1,5,6:-1,......}={1}/{1,0}
a:1->b:1
a:10->b:10
a:3,b:7->c:4,d:6
a:-1->a:-1
a,b:0.5,c:-0.3->d:1.2
{a:-1}{} ={b,b:-1},b:-1->a:-1，
aleph0+pi=aleph0。{a1,a2,a3,....}{a1,a2,a3,...,b:pi}。a1,a2,a3->b:3。a4->b:pi-3,a1:4-pi。a5->a1:pi-3,a2:4-pi。...a(n+4)->an:pi-3,a(n+1):4-pi。....
Aleph0*pi=aleph0。{a1:pi,a2:pi,a3:pi,....}{a1,a2,a3,...,}
a(6i-5),a(6i-4),a(6i-3)->a(i):pi-3,a(2i-1):6-pi。a(6i-2),a(6i-1),a(6i)->a(i):pi-3,a(2i):6-pi。
As you see, it proceeds with examples of set operations, when quantifiable cardinality is denoted with underscore. As I understand, sending me this proposal was one of the steps in realizing the idea of "Negative Cardlinaty", so, let this page be a place to add the follow up steps to achieve the wider verification and adoption of this concept.
負のカーディナリティは何かの不足であり、まだ実行されていないタスクです。それは、シーケンス、そして、そして時間と関係があります。本当に面白いです。
Negative cardinality is deficit of something, a task yet to be done. It has to do with sequence, then, and time. Really interesting.
カーディナリティ数は、質量の尺度になります。その場合、負のカーディナリティは負の質量です。 ????どういう意味ですか？
Cardinality number can be a measure of mass. Then negative cardinality is negative mass. ???? What does it mean?
//負のカーディナリティは負の質量ですか????
いいえ。カーディナリティはセット内の要素の数（いわゆる「セットのサイズ」）であるため、負のカーディナリティは0未満の要素を持つセットのサイズになります。
// negative cardinality is negative mass ????
No. Cardinality is the number of elements (so-called "size of the set") within a set, so, negative cardinality would be the size of the set that has less than 0 elements.
[skihappy]、負のカーディナリティで負の質量をモデル化できますが、概念としての負のカーディナリティは厳密には負の質量と同等ではないため、負の質量ではありません。
会計士にとって、負のカーディナリティは負の資産（負債）である可能性があり、他の専門家は他のドメインの概念である可能性があります。
[skihappy], you could model negative mass with negative cardinality, but negative cardinality as a concept is strictly is not equivalent to negative mass, so, it's not negative mass.
For an accountant, negative cardinality could be negative assets (liabilities), and other specialists it may be concepts in other domains.