Combination of n sets that produces a set of n-tuple

Question

Given n sets with $3$ elements: $X_i=\{a_i,b_i,c_i\}$ where $\{i\in\mathbb{N}\ |\ 1\leq i\leq n\}$. How can I define a n-tuple based on combination of this sets that produces the set $S$ with $3^n$ elements ($n$-tuples) as following: $S=\{(a_1,a_2,\cdots,a_n),(a_1,a_2,\cdots,b_n),(a_1,a_2,\cdots,c_n),\cdots,(c_1,c_2,\cdots,c_n)\}$.

Answer 1

If I understand your question correctly, what you want is simply the Cartesian product of the $n$ sets:

$$\prod_{i=1}^nX_i=\Big\{\langle x_1,\dots,x_n\rangle:x_1\in X_i,x_2\in X_2,\dots,x_n\in X_n\Big\}\;.$$