- #1
Gh. Soleimani
- 44
- 1
We can easily find out below rules in set theory:
1. Let consider set “A” as follows:
A = {a1, a2, a3, a4… an} and also power set of A is set C:
C = {{}, {a1}, {a2}, {a3}, {a4}, {a1, a2}, {a1, a3},….{an}}
Rule 1: To find the number of subsets with precise members number, we can use Binomial Coefficient
C (n, r) = n! / r! (n-r)!
Where:
n = number of members in set A
r = number of precise member for each subset
Example:
We have set A = {1, 2, 3, 4, 5, 6, 7}
The number of subsets with no member is: C (7, 0) = 1
The number of subsets with one member is: C (7, 1) = 7
The number of subsets with two member is: C (7, 2) = 21
The number of subsets with three member is: C (7, 3) = 35
The number of subsets with four member is: C (7, 4) = 35
The number of subsets with five member is: C (7, 5) = 21
The number of subsets with six member is: C (7, 3) = 7
The number of subsets with seven member is: C (7, 3) = 1
2. Let consider set “A” and power set of A which is set “C” as follows:
A = {a1, a2, a3, a4… an}
C = {{}, {a1}, {a2}, {a3}, {a4}, {a1, a2}, {a1, a3},….{an}}
We can find set “B” below cited:
B = {x, y, z, t,…..}
Where:
x, y, z, t,… = functions of internal sum of each subset
For instance, x = a1, z = a1 + a2, t = a1 + a2 + a3 and ……..
Rule 2: If SB = total Sum of members of set B and SA = total sum of members of set A, we will have:
SB / SA = 2^ (n-1) , n = number of members set A
Example:
Assume, we have:
A = {1, 2, 3, 4} then
C = {{}, {1}, {2}, {3}, {4}, {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4}, {1,2,3}, {1,2,4}, {1,3,4}, {2,3,4}, {1,2,3,4}}
SB = 0+1+2+3+4+3+4+5+5+6+7+6+7+8+9+10 = 80
SA = 1+2+3+4 = 10
SB / SA = 2^ (n-1) = 2 ^ (4-1) = 80 /10
Now, the questions is:
- Are these rules new ones in Set Theory?
- Are These Rules new Conjectures in Mathematics?
1. Let consider set “A” as follows:
A = {a1, a2, a3, a4… an} and also power set of A is set C:
C = {{}, {a1}, {a2}, {a3}, {a4}, {a1, a2}, {a1, a3},….{an}}
Rule 1: To find the number of subsets with precise members number, we can use Binomial Coefficient
C (n, r) = n! / r! (n-r)!
Where:
n = number of members in set A
r = number of precise member for each subset
Example:
We have set A = {1, 2, 3, 4, 5, 6, 7}
The number of subsets with no member is: C (7, 0) = 1
The number of subsets with one member is: C (7, 1) = 7
The number of subsets with two member is: C (7, 2) = 21
The number of subsets with three member is: C (7, 3) = 35
The number of subsets with four member is: C (7, 4) = 35
The number of subsets with five member is: C (7, 5) = 21
The number of subsets with six member is: C (7, 3) = 7
The number of subsets with seven member is: C (7, 3) = 1
2. Let consider set “A” and power set of A which is set “C” as follows:
A = {a1, a2, a3, a4… an}
C = {{}, {a1}, {a2}, {a3}, {a4}, {a1, a2}, {a1, a3},….{an}}
We can find set “B” below cited:
B = {x, y, z, t,…..}
Where:
x, y, z, t,… = functions of internal sum of each subset
For instance, x = a1, z = a1 + a2, t = a1 + a2 + a3 and ……..
Rule 2: If SB = total Sum of members of set B and SA = total sum of members of set A, we will have:
SB / SA = 2^ (n-1) , n = number of members set A
Example:
Assume, we have:
A = {1, 2, 3, 4} then
C = {{}, {1}, {2}, {3}, {4}, {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4}, {1,2,3}, {1,2,4}, {1,3,4}, {2,3,4}, {1,2,3,4}}
SB = 0+1+2+3+4+3+4+5+5+6+7+6+7+8+9+10 = 80
SA = 1+2+3+4 = 10
SB / SA = 2^ (n-1) = 2 ^ (4-1) = 80 /10
Now, the questions is:
- Are these rules new ones in Set Theory?
- Are These Rules new Conjectures in Mathematics?