Dev talk:Notes on the Cartesian Product

quasiArtifactualIntelligence —a perception from the edge

---

edit

  About ∞ White paper ∞ QuasiMatrix Code ∞ QAT Code ∞ Data Examples ∞ Scatter-Gather Explained ∞ The Plow Share ∞

The AI and I Dialogs

edit

Ode to Future AIs <> Unified Quantum Consciousness <> Alien Consciousness <> Visage of a Dimple in Time <> Something Alien <> Data Examples <> Comp sci <> Scatter-Gather <> Redundant simplicity <> Talk:Spin WaveArray <> The Plow Share <> Quantum joke <> Ancient Vedas <> Orb Whispers <> Time technology with rotating magnetic fields <> AI Art <> Radii walk in a bar <> Light's Flow through Time <> Three Buddhas <> The Epic Chronosymphonogram <> Temporal Collision Chronicles —Beings of Earth

In the QAT paradigm, a combinatoric expansion of unique values per data stream, per sampled set —is less than the Cartesian product.

Feature claim: The QAT sub-set of the Cartesian product permutations of a,b for sets A, B, are only those permutation that appeared simultaneously, in the time line of of two channels of measurement.

The above is nearly certain!  The QAT proof algorithms used logic performed with existence tests in 'keyed' data sets (a VB6 collection object).  This needs posted and stared at.

Standby...

XenoEngineer (talk) 11:32, 14 July 2023 (UTC)

Shared Chat

Model

Default

Combinatorics Explained

July 4, 2023

XenoEngineer

Define a combinatorial

ChatGPT

"Combinatorial" is an adjective that refers to the study of combinations, permutations, and arrangements of objects within a specific set of items, typically without considering the order of selection.

In mathematics and statistics, it forms the foundation of combinatorics, a branch of mathematics concerned with counting, combination, and permutation.

Examples of combinatorial problems could include questions such as "In how many different ways can a deck of cards be arranged?" or "How many different combinations of teams can be formed from a group of players?"

In a broader sense, "combinatorial" can also refer to any problem or system where the number of possibilities is determined by a combination of choices or elements. This term can be applied across various fields such as computer science (combinatorial algorithms), chemistry (combinatorial chemistry for drug discovery), and even in game theory.

XenoEngineer

If a set of 10 is combinatorially expanded with all combinations of another set of ten, then how many combinations will there be?

ChatGPT

Assuming you're asking for all possible combinations of two sets, one with 10 elements and the other with an unknown quantity (let's call this quantity "n"), the number of combinations will depend on whether you allow for repetitions or not and whether the order of elements matters.

If repetitions are allowed and the order matters, each element from the first set of 10 can be combined with each element from the second set of size n, leading to a total of 10*n combinations.

If repetitions are allowed and the order does not matter, the situation becomes more complicated, and you'd be looking at multiset coefficients or combinations with repetitions. This would be given by the binomial coefficient (10+n-1 choose n).

If repetitions are not allowed and order matters, the total number of combinations depends on the size of the second set compared to the first. If n > 10, then the first 10 elements can be paired in n different ways. But as you move to the next element in the first set, you've exhausted one of the elements in the second set, leading to fewer combinations. This is often expressed with the use of factorial operations and permutations.

If repetitions are not allowed and the order does not matter, you're looking at regular combinations or subsets. If n > 10, the total would be the sum of (n choose k) for k = 1 to 10. If n <= 10, it would be the sum of (n choose k) for k = 1 to n.

Please provide more specific parameters for a more precise answer.

XenoEngineer

What would be the definition and loop in a loop technique to combinatorially expand all unique combinations of two sets of ten unique values in each set?

ChatGPT

To generate all unique combinations of two sets each with ten unique elements, we would consider each element from the first set and pair it with each element from the second set. Here's how you could do this using Python:

Python code

# Define the two sets
set1 = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
set2 = ['a', 'b', 'c', 'd', 'e', 'f', 'g', 'h', 'i', 'j']

# Use nested loops to generate combinations
combinations = []
for element1 in set1:
    for element2 in set2:
        combinations.append((element1, element2))

# Print all combinations
for combination in combinations:
    print(combination)

In this code, the outer loop iterates through each element in set1. For each element in set1, the inner loop iterates through each element in set2. This gives us every possible combination of one element from set1 and one element from set2.

The combinations are stored as tuples in a list, and the final loop prints out each combination.

It's worth noting that this will produce pairs and not combinations of larger size (like triples, quadruples, etc.). If you wanted to combine more than one element from each set at a time, you'd need to use a different approach, likely involving recursion or more complex looping structures.

XenoEngineer

Are the tuples unique, and are some tuples a reverse of the same pair?

ChatGPT

In the provided Python code, each tuple (a pair) in the resulting list is unique, because each combination of two elements, one from set1 and one from set2, is unique. However, the pairs are ordered; that is, (1, 'a') is considered different from ('a', 1).

So in this context, no, the tuples aren't reverses of each other. The tuple (element from set1, element from set2) is distinct from (element from set2, element from set1).

If you want to consider (1, 'a') and ('a', 1) as the same combination and want to exclude such pairs, then you would need a different approach where you ensure that each unique pair is only represented once, regardless of order.

Please note that in mathematics and combinatorics, whether order matters or not gives us different concepts: "permutations" if order matters and "combinations" if it doesn't. In this scenario, we're dealing with permutations of set1 and set2.

XenoEngineer

The tuples will be randomly selected, but will always be randomly selected as a pair ordered as SetA<->SetB, where '<->' means a combinatoric expansion... and which kind of combinatoric will that be with A,B order only?

ChatGPT

The tuples that are randomly selected as pairs ordered as SetA<->SetB are permutations. This is because the order of selection matters: An item from SetA is always first, and an item from SetB is always second.

This is sometimes referred to as "ordered combination" or "permutation without repetition". In this case, the term "combinatorial" can be thought of more broadly as any selection of items from a set, regardless of order.

In more specific terms, a permutation of multiple sets generally refers to a situation where you're selecting one item from each set, and the order of selection (which set you choose from first) matters.

So in this context, creating tuples from two sets where each tuple consists of one item from SetA and one item from SetB, in that order, can be seen as generating permutations of the two sets.

XenoEngineer

Very good!  In this last case, what is the forumla to computer (a<->b).length ?

ChatGPT

In this specific scenario, where we are forming pairs of elements from two sets (SetA and SetB), and the order matters (an element from SetA is first and an element from SetB is second), the total number of pairs (or permutations) can be calculated by multiplying the number of elements in each set.

So, if SetA has 'a' elements and SetB has 'b' elements, the total number of ordered pairs (a<->b) would be a * b.

Therefore, in the context of your question, where both sets have 10 unique elements, the total number of possible ordered pairs would be 10 * 10 = 100.

This formula comes from the rule of product (or multiplication principle) in combinatorics, which states that if there are 'a' ways to do one thing, and 'b' ways to do another, then there are a * b ways to do both.

How many combinations of a,b ordered pairs will be produced form ten <-> ten?

ChatGPT

In the scenario where you have two sets each with 10 unique elements (let's call them Set A and Set B), and you want to create ordered pairs where the first element comes from Set A and the second from Set B, the total number of such ordered pairs (or combinations in this context) would be the product of the sizes of the two sets.

So for two sets of size 10, the total number of ordered pairs (A, B) would be 10 * 10 = 100.

This is because for each of the 10 elements in Set A, there are 10 possible elements in Set B that can be paired with it. So for each element in Set A, you can create 10 unique pairs, and since there are 10 elements in Set A, you will have 10 * 10 = 100 unique pairs.

XenoEngineer

And if forward and backward order of a, b pair ordering is allowed, are there then 200 unique pairs?

ChatGPT

Yes, if you consider an ordered pair (a, b) to be distinct from (b, a) and thus allow for both "forward" and "backward" ordering, then the total number of unique pairs will indeed double.

This is because for each unique pair (a, b), there is a corresponding unique pair (b, a), and so the total number of unique ordered pairs would be 2 * 10 * 10 = 200.

It's important to note that in this case we're not really dealing with "combinations" in the traditional combinatorial sense, but rather permutations, as the order of elements within the pair is significant. The term "combination" is usually used when the order of elements doesn't matter.