Problems: Mathematics.
Solutions: Negative Cardinality.

# Quantitative Cardinality Sets Project P&L: 0 ħ (≃ 0 USD)

Brining sets with quantifiabiable cardinality into common use in mathematics.
YAML Project

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.

[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.

// 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.

Cardinality number can be a measure of mass. Then negative cardinality is negative mass. ???? What does it mean?

Negative cardinality is deficit of something, a task yet to be done. It has to do with sequence, then, and time. Really interesting.

    :  --
: Mindey
:  --


skihappy,