I have come across the question: Combination of n sets that produces a set of $n$-tuple.
Given two ordered sets with $n=3$ elements: $A=\{2, 4, 6\}$, $B=\{1, 3, 5\}$, $a_i < b_i$, $i=1, 2, \ldots, n$.
Question. How to name and define the set based on combination of this sets that produces the set $C$ with $2^n$ elements ($n$-tuples) as following: $$C=\left\{\\(1,3,5),\\ (1,3,6),\\(1,4,5),\\(1,4,6),\\(2,3,5),\\(2,3,6),\\(2,4,5),\\(2,4,6)\\\right\}.$$
My attempt is:
Name: $C$ is the set of ordered permutation without replacement of two $n$-elements sets $A$ and $B$.
Update.
Definition: $C=\{(x_1, x_2,\dots, x_n): x_i =a_i \vee b_i, i=1,2, \ldots, n\}=\{c_1, c_2, \ldots, c_N\}$, where $N=2^n$.
I'm confused because in my definition because the sign $\vee$ denotes the logical OR there, but elements of $A$ and $B$ are integers.
I have read here:
Logical operators are typically used with Boolean (logical) values. When they are, they return a Boolean value. However, the && and || operators actually return the value of one of the specified operands, so if these operators are used with non-Boolean values, they may return a non-Boolean value.