vb.net Pair Combinations to Create All Possible Sets - vb.net

Question

I need help figuring out how to go about programming this problem. I have an unknown number of pairs. Each pair is a Length x Width. I want to create sets of every possible combination of either a Length or Width from each pair. Here is an example of what I'm trying to do:
If I input 3 pairs, (1x2) (3x4) (5x6) I would get the following sets:
(1,3,5) (1,3,6) (1,4,5) (1,4,6) (2,3,5) (2,3,6) (2,4,5) (2,4,6)
So if I had 4 pairs, it would create a total of 16 sets, etc. I need to be able to input each pair and after all pairs have been entered, I need it to print out the sets. It can never include both numbers from a given pair in the same set. How do I create a loop or is there a built in math function that could produce all possible sets from a given number of pair inputs? I hope I described the problem well enough but if not, please ask questions. Thanks

Answer 1

This is called Cartesian product.
For example, if we have two sets A and B, such that
A = {1,2}
B = {3,4}
Then the result of the Cartesian product A x B is equal to
A x B = {(1,3),(1,4),(2,3),(2,4)}
If now we want to make the Cartesian product between the result obtained above and a new set, for example:
N = {5,6}
The result of the Cartesian product A x B x N, is equal to
A x B = {(1,3),(1,4),(2,3),(2,4)}
N = {5,6}
──────────────────────────────────────────────────
A x B x N = {(1,3,5),(1,3,6),(1,4,5),(1,4,6),(2,3,5),(2,3,6),(2,4,5),(2,4,6)}
Each element of the first set must be paired with each element of the second set.
I have developed 4 solutions to the Cartesian product:
Using a mathematical model, but without recursion. This solution
vectors using calculating for each combination number.
Using recursion, with the Collections class.
using the List (Of ...) class, also with recursion.
These three solutions seemed to me difficult to explain to you.
Furthermore, it is very difficult for me to explain my thoughts in English, because my native language is Castilian.
So I made the effort to create another solution that does not use recursion, which was more simple and friendly for any programmer.
Finally, I could create a satisfactory solution. Easy to understand and without recursion.
It is also very versatile. Any number of sets is accepted, as required, from 2 onwards.
You can also use any number of items. This depends on the requirements of each developer.
I hope this 4th. solution I devised, will please you, esteemed colleagues.
Only need a ListBox1 within Form4. Here it is:
Public Class Form4
Private Sub Form4_Load(sender As System.Object, e As System.EventArgs) Handles MyBase.Load
'┌─────────── Temporary code for example ───────────┐
Dim Set_1 As List(Of String) = New List(Of String)
Dim Set_2 As List(Of String) = New List(Of String)
Dim Set_3 As List(Of String) = New List(Of String)
Set_1.Add("1")
Set_1.Add("2")
Set_2.Add("3")
Set_2.Add("4")
Set_3.Add("5")
Set_3.Add("6")
'└──────────────────────────────────────────────────┘
Dim Sets As List(Of Object) = New List(Of Object)
Sets.Add(Set_1)
Sets.Add(Set_2)
Sets.Add(Set_3)
Dim product As List(Of String) = Sets(0)
For i = 1 To Sets.Count - 1
product = CartesianProduct(product, Sets(i))
Next
For Each element As String In product
Me.ListBox1.Items.Add(element)
Next
End Sub
Private Function CartesianProduct(ByVal Set_A As List(Of String), ByVal Set_B As List(Of String)) As List(Of String)
Dim product As List(Of String) = New List(Of String)
For Each a As String In Set_A
For Each b As String In Set_B
product.Add(a & b)
Next
Next
Return product
End Function
End Class
Have a nice day! :)