Let us say we have 2 tuples whose elements are all distinct (relative to the given tuple they are apart of) and have some ordering relation already defined on them, given as:
$$t_1 =(\psi_1, \psi_2, \psi_3,\ldots,\psi_n)$$
and
$t_2 = (\alpha_1, \alpha_2, \alpha_3,\ldots,\alpha_m)$
and we wish (for whatever reason) to have a binary operation $*$ such that $t_1 * t_2$ yields a tuple $t_3$ which is such that:
1) There is no element from either $t_1$ or $t_2$ which is not in $t_3$
2) $t_3$ does not contain duplicate elements, so that each element of $t_3$ is distinct
3) The order of the elements in $t_3$ is that of the defined ordering relation that needs to be given.
So, for example:
Let $t_1 = (3,7,8,1)$
and $t_2 = (7,4,2,5,9)$
And let the relation of the elements that the operation of $*$ will go by, be the standard less-than relation.
So then we would have:
$t_1 * t_2 = t_3 = (1,2,3,4,5,7,8,9)$
(note the lack of '$7$' appearing twice, as well as the fact that the elements are ordered according to $<$)
How would one be able to define this operation with just symbols or would defining $*$ via the rules given above be acceptable?
Thanks
edit: to get more exposure I'm adding a set-theory tag, since tuples are after all defined as (ordered) sets.
Best Answer
A discussion occurred between us in chat, so here I will highlight the end results.
Depending on how exactly you define a tuple, there are a number of options, but the end result is that the operation is essentially the set of all elements appearing in at least one of $t_1$ or $t_2$ placed into a tuple with entries in the same order as they would appear in the underlying set.
Interpreting tuples as ordered sets:
Letting $t_1 = (A,\prec_a)$ and $t_2 = (B,\prec_b)$ and the underlying set be $(X,\prec_x)$ of which both $A$ and $B$ are subsets of $X$, we have:
$$t_1*t_2 = \left((A\cup B), \prec_x\cap (A\cup B)^2\right)$$
keeping in mind that an order on $X$ is itself a set and is a subset of the cartesian product of $X$ with itself. The intersection them makes sense. Alternatively, one could just refer to the resulting order for $t_1*t_2$ as the inherited order, for which various notations exist such as $\prec_x|_{A\cup B}$.
Interpreting tuples as injective functions from $[n]$ to $X$:
Letting $(X,\prec)$ be the underlying ordered set and letting $t_1=f_a$ where $f_a\in X^{\underline{[n_a]}}$ (i.e. letting $f_a$ be an injective function from $[n_a]$ to $X$) and $t_2=f_b\in X^{\underline{[n_b]}}$, we have:
$$t_1*t_2 = g\in (f_a(A)\cup f_b(B))^{[|f_a(A)\cup f_b(B)|]}~\text{satisfying}~\\\forall j,k\in[|f_a(A)\cup f_b(B)|]~\text{one has}~j<k\implies g(j)\prec g(k)$$
Both of these seem like a lot of words to use for something that can be described much more simply with english rather than symbols: $t_1*t_2$ is the unique tuple such that each entry of $t_1$ and $t_2$ appear once and they appear in the same order as the underlying set.