I've read from the theory of sets that the definition of ordered pair is foundational. To define formally the term of $n$-tuple I suppose we need to use the concept of ordered pair as well as the definition of recursion. So I was wondering about the difference between doing so and defining an $n$-tuple just like we do with matrices, I mean we can just say that an $n$-tuple is a sequence (a function with domain $\{ 1,2,3,4,…,n\}$). What is the advantage of making a definition by recursion? What is the difference between a sequence and an $n$-tuple? What is the difference between considering a vector as a matrix and considering a vector as $n$-tuple?
Linear Algebra – Definition of n-Tuple Explained
definitionelementary-set-theorylinear algebra
Best Answer
Just a remark on your statement the definition of ordered pair is foundational.
Do you mean an ordered pair is defined axiomatically? Anyway, a way to define an ordered pair, which I learned from Halmos' Naive Set Theory, is by $$ (a, b) := \{ \{ a \}, \{ a, b \} \}. $$ It has all the properties you desire of an ordered pair, plus some incidental ones you may safely disregard.