What does the Cartesian product really mean. It is said that the Cartesian product of two sets $A$ and $B$ means the set of all ordered pairs $(a,b)$, where $a\in A$ and $b \in B$. But what is the use of defining it?

For what purpose it is defined in early days? What is the use of Cartesian product in defining a relation and a function. I understand clearly the meaning of function but feel difficulty in understanding the basis in which Cartesian product is defined.


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    $\begingroup$ But what does the pair of elements that are related mean? $\endgroup$ – user355797 Jul 24 '16 at 16:39
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    $\begingroup$ In what way does it related $\endgroup$ – user355797 Jul 24 '16 at 16:40
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    $\begingroup$ For example, the relation "less-than" on real numbers relates certain real numbers to each other. $3<5$, thus $(3,5)\in L$ where $L$ denotes the less-than relation. $\endgroup$ – celtschk Jul 24 '16 at 16:44
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    $\begingroup$ Nothing has to be applicable or motivated by reality, or even by mathematical convenience. You can just define whatever you want, it's just a definition. $\endgroup$ – Ovi Jul 24 '16 at 17:19
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    $\begingroup$ in early time, Descartes used it to describe a plane with coordinates $\endgroup$ – user354674 Jul 24 '16 at 18:31