Are relations with different attribute names union-compatible?

Question

If I have two relations:

$$\begin{align} R_1(Name: string, Address: string)\\ R_2(Title: string, Location: string) \end{align}$$

They are of the same arity, and their corresponding attributes belong to the same domain. However, their attribute names are different.

Are they union-compatible?

If they are, then what would be the attribute names of, say, their union?

Answer 1

Well, I found answers to my question in these two resources:

1. Week 3 – Relational Algebra (PDF)
2. 4. Relational Operators I - Relational Theory for Computer Professionals

In summary, the union of two relations is not valid unless the relations, in addition to having the same arity and attribute domains, have the same attribute names. In fact, the second resource (cited above) mentions:

The reason it doesn’t work is that UNION requires its operands to be of the same type, meaning that corresponding attributes have to be of the same type and have the same name. (Formally, in fact, they have to be the very same attribute.)

If you want to perform union on such relations, you need to rename their attributes, so that they have the same attribute names.

It is worth mentioning that the order in which the attribute names are specified does not matter. That is, if you have a relation, say, $R_1(Name, Title)$, you can also write it as $R_1(Title, Name)$. It follows that if $R_1(Name, Title)$ and $R_2(Title, Name)$ are two relations of the same type, then you do not need to rename to perform their union.