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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?