I am a bit of a newbie in math and I am struggling with the question written below.
Question: Let $(X_1,\leq_{X_1}),(X_2,\leq_{X_2})$ to be a well-ordered set, and $X_1\cap X_2=\emptyset$
On $X_1\cup X_2$, define a well-ordered set preserving the order relations defined set on $X_1$ and $X_2$ sets.
I am not very familiar with the notion of "well-ordered set", I have read it but I do not seem to grasp what role it would play to answer the question. Can you give me some insights on how to use it and on what does it mean to preserve teh property for the union? What properties does the realtion between an element of $X_1$ and $X_2$ should have? Am I allowed to arbitrary sort elements of $X_1$ with respect to those of $X_2$ as I wish as far as I preserve the order for the elements within the original sets?
Update on the answer: Thanks to the comments and answer I understood that the question was rather trivial. Indeed having the freedom to define the relation we can simply do as follows: preserve the relation we have within the sets $X_1$ and $X_2$, and define a suitable behaviour for the remaining case. A simple example would be to use the set name to order elements of different sets in an arbitrary way. This is feasible since we have an empty intersection. Specifically, a solution would be to define $\leq_{X_1\cup X_2}$ on $X_1\cup X_2$ as follows, which also generalize to more disjuncted sets:
\begin{equation} u \leq_{X_1\cup X_2} v = \begin{cases} u \leq_{X_i} v & \text{if}\ u\in X_i \wedge v \in X_i \\ i \lt j & \text{if}\ u\in X_i \wedge v \in X_j \wedge i \neq j \end{cases} \end{equation}
A well-ordered set $(X,\leq_X)$ is a set $X$ equipped with a relation $\leq_X$ such that:
$$1. \forall x\in X,\, x\leq_Xx\,\,\,\,\,\,\text{(Reflexive)}\\ 2. \forall x,y\in X, \, x\leq_Xy\,\, \text{and} \,\, y\leq_Xx\Rightarrow y=x\,\,\,\,\,\,\text{(Antisymmetric)}\\ 3. \forall x,y,z\in X, \, x\leq_Xy\,\, \text{and} \,\, y\leq_Xz\Rightarrow x\leq_Xz\,\,\,\,\,\,\text{(Transitive)}\\ 4. \forall x,y\in X, \text{ either }x\leq_Xy\text{ or } y\leq_Xx\,\,\,\,\,\,\text{(Connex)} $$
Given that $X_1$ and $X_2$ are disjoint in your problem, can you think of an order $\leq_{X_1\cup X_2}$ which makes $(X,\leq_{X_1\cup X_2})$ into a well-ordered set? (Hint: try "putting $X_2$ after $X_1$")
You might want to look into partial orders, and ordinals.