Is this sentence complex or compound:
I have not only a book but also a computer.
Here, I think two independent clauses have been connected by the conjunction but. So, as per my idea, it's a compound sentence? What do you think?
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The sentence is simple; it contains only one clause. Let's confirm this by removing the not only...but also part:
I have a book and a computer.
In the sentence above, I have removed the emphasis of having a computer in addition to having a book, but the basic structure is the same. The sentence contains just one clause made up of a subject (I) and a predicate (have a book and a computer).
Though it is not a compound sentence, your sentence does contain a compound object consisting of the nouns book and computer, which are connected by the correlative conjunction not only...but also.
Though not only X but Y does not occur across two separate clauses in the sentence you gave, it can do so:
Not only was he short, (but) he was also fat.
(Per CGEL, but can be omitted when it's introducing the main clause.)
You suggest that you think this is a compound sentence. If so, there would have to be two independent clauses joined by a conjunction. You suggest that "but" is the conjunction. If that were true, the clauses would be "I have not only a book" and "also a computer". The second phrase does not have a verb so it cannot be an independent clause. The first sentence has both a subject and a verb, but it doesn't make sense by itself due to the "not only". So this is not a compound sentence.
So is this a complex sentence? To be a complex sentence, you need an independent clause and a dependent clause joined by a subordinating conjunction or a relative pronoun. In this case, we do not have a subordinating conjuction or a relative pronoun, so it cannot be a complex sentence.
The correct answer is that this is a simple sentence with a compound object (sorry about the video link, I couldn't find a good text description). The subject is "I", the verb is "have" and the object is the compound direct object "not only a book but also a computer". The phrases "not only... but also..." make up a correlative conjunction. The fact that this is a simple sentence can be seen more easily if you change it to "I have a book and a computer" or "I have a book". In those two sentences and the original, the only thing that is changed is the object ("not only a book but also a computer", "a book and a computer", and "a book"); the structure of the sentence is the same.