Mathematical connection in the cartesian product

In summary, the Cartesian product is a mathematical connection between the elements of two sets, A and B, that gives more information than a set with cardinality |A||B|. It also comes with canonical projections that satisfy specific mapping properties. While it is theoretically possible to think of the elements of A and B as contained in the elements of the Cartesian product, there is no practical reason to do so.
  • #1
V0ODO0CH1LD
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mathematical "connection" in the cartesian product

What is the mathematical connection between elements of a cartesian product ##A\times{}B## and the elements of the sets ##A## and ##B##?

In other words, what is the difference between the set ##A\times{}B## and just any set ##Z## with ##|A|.|B|## elements that creates no contradictions if I choose to make a connection in my head between each element of it and one element of ##A## and one element of ##B## the same way the cartesian product of ##A## and ##B## does?

Because if you forget the usual notation for elements of a cartesian product (e.g. ##(a,b)##, ##a\times{}b##, ##ab##) all you have is a set with cardinality ##|A|.|B|## that you as a human connect to the elements of the sets involved in the product in a particular way, usually through notation.

But if I have ##|Z|=|A|.|B|## how can I prove or disprove that the set ##Z## is the cartesian product of ##A## and ##B##? Are the elements of ##A## and ##B## set theoretically contained in the elements of ##A\times{}B##?

If I choose to say the set ##\{1,2,3,4\}## is the cartesian product of the sets ##\{a,b\}## and ##\{c,d\}## is that incorrect just because of the way that I chose to write the sets down?
 
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  • #2
The Cartesian product of sets X0 and X1 actually gives you more information than a set with cardinality |X0||X1|. It also comes equipped with canonical projections pi:X0×X1→Xi that satisfy nice mapping properties. Any set Z equipped with projections qi:Z→Xi satisfying these same properties could equally well be called the product of X0 and X1 and in some cases this identification is made. On the other hand, if someone asks whether {1,2,3,4} is the product of {a,b} and {c,d}, then you should probably answer "no" since point-wise these sets differ.

There are other reasons to pay attention to the usual description of Cartesian products by the way. When you have sets with structure (like groups, rings, topological spaces, etc.) often times a product of these objects is constructed by looking at the usual Cartesian product of the underlying sets and then defining the extra structure on that. So it genuinely makes life easier to keep the ordered pair definition in mind.
 
  • #3
I don't know why it occurred to me to google "cartesian product" only after I posted my question, but anyway..

In ZFC everything is a set, which means every element of every set is a set. So even though the ordered pair (a,b) has been defined as (a,b) = {{a},{a,b}} and (a,b) = {{a},ø},{{b}}} for different reasons, the only thing that matters is that your definition makes sure that (a,b) = (c,d) <=> a = b and c = d. In a sense it is okay to think that the element (a,b) in AxB set theoretically contains the elements a in A and b in B.
 
  • #4
V0ODO0CH1LD said:
So even though the ordered pair (a,b) has been defined as (a,b) = {{a},{a,b}} and (a,b) = {{a},ø},{{b}}} for different reasons, the only thing that matters is that your definition makes sure that (a,b) = (c,d) <=> a = b and c = d.

This is true for defining ordered pairs. For Cartesian products the important thing is the canonical projections.

In a sense it is okay to think that the element (a,b) in AxB set theoretically contains the elements a in A and b in B.

You hypothetically can but there really is no reason to.
 
  • #5
Or is there a fundamental mathematical connection between the elements of ##A\times{}B## and the elements of ##A## and ##B## that cannot be replicated in any other set with the same cardinality?

The mathematical connection in the cartesian product lies in the way that the elements of the set ##A\times{}B## are constructed. Unlike a general set with the same cardinality, the elements of the cartesian product are ordered pairs, where the first element comes from the set ##A## and the second element comes from the set ##B##. This ordering and pairing of elements is what creates a unique mathematical connection between the elements of the cartesian product and the elements of the sets ##A## and ##B##.

In your example, the set ##\{1,2,3,4\}## cannot be considered the cartesian product of the sets ##\{a,b\}## and ##\{c,d\}## because it does not follow the ordering and pairing of elements that is characteristic of a cartesian product. The elements of the cartesian product must have a specific structure and relationship to the elements of the sets involved, which cannot be replicated in any other set with the same cardinality.

Furthermore, the elements of the sets ##A## and ##B## are not set theoretically contained in the elements of the cartesian product. They are distinct sets with their own elements, and the cartesian product simply combines these elements in a specific way to create a new set with a unique mathematical connection.
 

Related to Mathematical connection in the cartesian product

1. What is the cartesian product?

The cartesian product is a mathematical operation that takes two sets and creates a new set containing all possible ordered pairs of elements from the original sets. It is denoted by A x B, where A and B are the original sets.

2. How is the cartesian product related to mathematical connections?

The cartesian product allows us to find connections between two sets by creating a new set that contains all possible combinations of elements from the original sets. This can help us identify patterns and relationships between the elements in the two sets.

3. Can the cartesian product be applied to more than two sets?

Yes, the cartesian product can be applied to any number of sets. For example, the cartesian product of three sets A, B, and C would be denoted by A x B x C and would create a new set containing all possible ordered triples of elements from the three original sets.

4. How is the cartesian product different from other mathematical operations?

The cartesian product is different from other operations, such as addition or multiplication, because it does not combine or manipulate the elements in any way. It simply creates a new set containing all possible combinations of elements from the original sets.

5. What is the significance of the cartesian product in mathematics?

The cartesian product is an important concept in mathematics because it allows us to explore and understand the relationships between different sets. It is also used in various fields, such as computer science and statistics, to represent and analyze data sets and their connections.

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