Cartesian products

A Cartesian product consists of ordered pairs with one component from each set.

Different ways to understand it

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Relations
A binary relation is a subset of a Cartesian product; its properties constrain related pairs.
Set operations
Union, intersection, difference and complement combine sets relative to a specified universe.
Axiomatic systems
An axiomatic theory derives theorems from stated axioms and inference rules.
Bijections
A bijection is both injective and surjective and has a two-sided inverse.
Countability
A set is countable if it is finite or can be enumerated by natural numbers.