I was proving that the cardinality of Cartesian product of m(m>1) non-empty finite sets is the product of cardinalities of the m sets. it can be easily proved using the fundamental principle of counting(rule of product)-
this is somewhat similar to -
But I felt the need of proving one more point in order to complete the proof of the above stated theorem- "Each and every ordered n-tuple so formed is unique." If it hadn't been so, the cardinality of the Cartesian product(which itself is a set),may be less than product of cardinalities of individual sets {since duplicate elements are not mentioned in a set}
I am able to understand it by intuition, but am not able to prove it in formal language. please help.
p.s. consider the Cartesian product of 2 sets
When we enumerate $P$ as $\{x_1,\ldots,x_m\}$ and $Q$ as $\{y_1,\ldots,y_n\}$, it’s tacitly understood that the elements $x_1,\ldots,x_m$ are distinct, and the elements $y_1,\ldots,y_n$ are distinct. (Of course any of the $x_i$ may be equal to any of the $y_i$.) Suppose that $\langle x_i,y_j\rangle=\langle x_k,y_\ell\rangle$. Then by the definition of ordered pair we know that $x_i=x_k$ and $y_j=y_\ell$, and therefore $i=k$ and $j=\ell$. Thus, all $mn$ ordered pairs are distinct.