Final answer:
There are O(2^n) ways to join n relations, which represents an exponential growth in the number of combinations as new relations are added. This complexity is due to each relation potentially being joined with every other relation.
Step-by-step explanation:
The question you're asking relates to the complexity of joining n relations, which is a concept in the area of databases and computer science that can also be understood through mathematical principles. When we talk about joining n relations, we are considering the different ways these relations can be combined. The complexity of this operation is not O(n), O(n log n), or O(n^2), but rather O(2^n). This is because each relation can potentially be joined with every other relation, and the number of combinations grows exponentially with the number of relations.
To illustrate this with an example, if you have 3 relations, there are 7 different ways to join them (excluding the option of not joining any relations): A with B, A with C, B with C, A with B and C, etc. As you can see, adding just one more relation greatly increases the number of possible combinations (joins), hence the complexity being of the order of 2 raised to the power of n, which is O(2^n). Therefore, the correct answer to the question is D) O(2^n).