Pairwise disjoint mathematical proof

Set 8 = {{3, 9}, {9, 4}, {3, 9, 4}, {4, 9, 1}, {4, 9, 2}, {4, 9, 5}, {4, 9, 6}, {4, 9, 7}, {4, 9, 8}, {5, 9, 10}};Set 9 = {{3, 10}, {10, 4}, {3, 10, 4}, {4, 10, 1}, {4, 10, 2}, {4, 10, 5
  • #1
vs140580
3
0
Set 1 {{1, 2}, {2, 3}, {1, 2, 3}, {2, 3, 4}, {2, 3, 5}, . . . , {2, 3, n}};
Set 2 {{3, 5}, {5, 4}, {3, 5, 4}, {4, 5, 1}, {4, 5, 2},{3, 5, 6}, {3, 5, 7}, . . . , {3, 5, n}};
Set 3 {{3, 1}, {1, 4}, {3, 1, 4}, {1, 4, 2}, {1, 3, 5}, {1, 3, 6}, . . . , {1, 3, n}};
Set 4 {{1, 5}, {5, 2}, {1, 5, 2}, {2, 4}, {4, 3}, {5, 6}, {5, 7}, . . . , {5, n}};
Set 5 {{3, 6}, {6, 4}, {3, 6, 4}, {4, 6, 1}, {4, 6, 2}, {4, 6, 5},{5, 6, 7}, {5, 6, 8}, . . . , {5, 6, n}};
Set 6 for n = 8,
{{3, 7}, {7, 4}, {3, 7, 4}, {4, 7, 1}, {4, 7, 2}, {4, 7, 5},{4, 7, 6}, {5, 7, 8}},
and for n ≥ 9, {{3, 7}, {7, 4}, {3, 7, 4}, {4, 7, 1}, {4, 7, 2}, {4, 7, 5}, {4, 7, 6},{5, 7, 8}, {5, 7, 9}, . . . , {5, 7, n}};
for n ≥ 11 and for k ∈ {7, . . . , n − 4},
set K: {{3, k + 1}, {k + 1, 4}, {3, k + 1, 4},{4, k + 1, 1}, {4, k + 1, 2},{4, k + 1, 5}, {4, k + 1, 6},
. . . , {4, k + 1, k},{5, k + 1, k + 2}, {5, k + 1, k + 3}, . . . , {5, k + 1, n}};
for n ≥ 10,
set n-3: {{3, n − 2}, {n − 2, 4}, {3, n − 2, 4}, {4, n − 2, 1}, {4, n − 2, 2},
{4, n − 2, 5}, {4, n − 2, 6}, . . . , {4, n − 2, n − 3},
{5, n − 2, n − 1}, {5, n − 2, n}};
for n ≥ 9,
set n-2 : {{3, n − 1}, {n − 1, 4}, {3, n − 1, 4}, {4, n − 1, 1}, {4, n − 1, 2},
{4, n − 1, 5}, {4, n − 1, 6}, . . . , {4, n − 1, n − 2}, {5, n − 1, n}};
Set n-1 : {{3, n}, {n, 4}, {3, n, 4}, {4, n, 1}, {4, n, 2},
{4, n, 5}, {4, n, 6}, . . . , {4, n, n − 1}}.

How to prove the n−1 sets are pairwise disjoint in general

I could see it by placing a value for n greater than 8.
 
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  • #2
For example, let us take n = 10. Set 1 = {{1, 2}, {2, 3}, {1, 2, 3}, {2, 3, 4}, {2, 3, 5}, {2, 3, 6}, {2, 3, 7}, {2, 3, 8}, {2, 3, 9}, {2, 3, 10}};Set 2 = {{3, 5}, {5, 4}, {3, 5, 4}, {4, 5, 1}, {4, 5, 2}, {3, 5, 6}, {3, 5, 7}, {3, 5, 8}, {3, 5, 9}, {3, 5, 10}};Set 3 = {{3, 1}, {1, 4}, {3, 1, 4}, {1, 4, 2}, {1, 3, 5}, {1, 3, 6}, {1, 3, 7}, {1, 3, 8}, {1, 3, 9}, {1, 3, 10}};Set 4 = {{1, 5}, {5, 2}, {1, 5, 2}, {2, 4}, {4, 3}, {5, 6}, {5, 7}, {5, 8}, {5, 9}, {5, 10}};Set 5 = {{3, 6}, {6, 4}, {3, 6, 4}, {4, 6, 1}, {4, 6, 2}, {4, 6, 5}, {5, 6, 7}, {5, 6, 8}, {5, 6, 9}, {5, 6, 10}};Set 6 = {{3, 7}, {7, 4}, {3, 7, 4}, {4, 7, 1}, {4, 7, 2}, {4, 7, 5}, {4, 7, 6}, {5, 7, 8}, {5, 7, 9}, {5, 7, 10}};Set 7 = {{3, 8}, {8, 4}, {3, 8, 4}, {4, 8, 1}, {4, 8, 2}, {4, 8, 5}, {4, 8, 6}, {4, 8, 7}, {5, 8, 9},
 

Related to Pairwise disjoint mathematical proof

What is a pairwise disjoint mathematical proof?

A pairwise disjoint mathematical proof is a type of proof used in mathematics to demonstrate that two or more sets are disjoint, meaning they have no elements in common. It shows that the intersection of the sets is empty, and thus they do not overlap in any way.

How is a pairwise disjoint mathematical proof different from other types of proofs?

Unlike other types of proofs that focus on proving the existence or uniqueness of a solution, a pairwise disjoint proof is concerned with proving that two or more sets have no elements in common. It is often used to prove statements about sets or functions.

What are some common strategies used in a pairwise disjoint mathematical proof?

One common strategy is to use a proof by contradiction, assuming that the sets are not disjoint and then showing that this leads to a contradiction. Another approach is to use a direct proof, where each element in one set is shown to be distinct from every element in the other set.

Can a pairwise disjoint mathematical proof be used for infinite sets?

Yes, a pairwise disjoint proof can be used for infinite sets as well. In this case, the proof may involve showing that the sets have no elements in common for any value in a given range, rather than directly comparing individual elements.

How is a pairwise disjoint mathematical proof used in other fields of science?

Pairwise disjoint proofs are used in various fields of science, such as computer science, physics, and statistics. They are often used to prove the properties of sets, functions, and other mathematical structures that are essential for solving problems and making predictions in these fields.

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