Prof. Lee wrote the preface carelessly, and so did Prof. Kim carefully.

That is an example sentence for the sample answer.

The answer says The verb 'write' has the NP 'the preface' as a complement, ~~~~~

1. My question is can "the" underlined come in the answer?

In the example sentence, there is only a past form(wrote) of a verb write; then how "the" can be used?

2. Shouldn't " <

be used instead of ' <

=? eg)"abc", 'abc' ;

Again, there is only a past form (wrote) of a verb write. So putting '<= is wrong because a normal form of a verb write can not be found in the example sentence as a quatation
Sorry, but I have no idea what you are talking about, moon.

The original sentence about Prof Lee and Prof Kim is incorrectly composed. Also, 'preface' in that sentence is the direct object of 'write', not a complement.

The rest of your post is inscrutable. Could you please explain in a different way?
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Mister Micawber's right about the first sentence being incorrectly composed. If you're learning from a guide that presents such a sentence as an example of good English, choose a different guide!

Concerning your questions:

1. The underlined 'the' is referring to a specific instance of the verb 'write'; therefore, the definite article is the correct choice. However, in the answer, 'write' should really be 'wrote'. One could argue that 'wrote' is just an inflection of the verb 'write' and that 'write' is, therefore, still correct. But using 'wrote' would be clearer.

2. Using single quotation marks (') rather than double quotation marks (") is still good English - provided you're consistent in their use, i.e. if you've used single quotation marks in a document, don't use double quotation marks elsewhere in the same document (except in the case of a quotation within a quotation). Double quotation marks are standard in the USA. Single quotation marks are standard in the UK; however, some popular UK publications use double quotation marks. Here in Australia, single quotation marks are the old standard, but both systems are in wide use. Which system you use is a question of style not correctness. Choose double quotes if you're writing for a US audience and single if you're writing for a UK audience.

Concerning the use of the word complement:

The term complement is reserved by some grammarians (including Mister Micawber and I) for the item following a copular verb (e.g. He is white - 'is' being the copular verb, 'white' the complement). However, 'others apply it to any item which completes the verb phrase: objects, adverbs, verb phrases or complements (as just defined).' Those are the words of Pam Peters, the author of The Cambridge Guide to English Usage.

Therefore, the answer Moon quoted is not wrong; it just isn't as precise as some of us would like it to be.

P.S. Copular verbs are also known as linking verbs, connecting verbs, copulas, copulative verbs, and equational verbs.
moon7296Again, there is only a past form (wrote) of a verb write. So putting '<= is wrong because a normal form of a verb write can not be found in the example sentence as a quatation
Quotation marks are not used solely for indicating direct quotations. They are also used in four other situations: (1) when you mean so-called-but-not-really <e.g. if he's a 'champion', he certainly doesn't act like one>; (2) when you're coining a new word for something; (3) when you're marking titles of TV and radio programs, magazine and newspaper articles, book chapters, poems, short stories, and songs; and (4) when you're referring to a word as a word, such as in your example, i.e. you're not talking about the act of writing, you're talking about 'wrote' being a verb.

Some of America's highest authorities on English usage suggest that you use italics when you're referring to a word as a word, but quote marks are still acceptable and are often used for this purpose.
Thank you so much; I was going to rewrite my question after Mr.Micauber said some part of my post is inscrutable. But you caught all the exact points I want to know and added detailed about the question through your 2 answers. I feel so thankful again; feeling the same as my previous question.

Could I ask one more question?

1. you used 'i.e' in "i.e. you're not talking about the act of writing, you're talking about 'wrote' being a verb".

2. What is the difference between e.g and i.e ?

and what is cf <=?? (that is probably similar to e.g; but I don't know exactly what it is)
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i.e. is an abbreviation of the Latin phrase id est. It means 'that is'. You use it before rearticulating a statement that you just made. 'In other words' is another way of saying 'i.e.'

e.g. is an abbreviation of the Latin phrase exempli gratia. It means 'for example'. You use it before providing an example or examples that illustrate the statement you just made.

cf. is an abbreviation of the Latin word confer. It means 'compare with'. You use it to draw the reader's attention to a related concept or to another passage of your document where you've discussed a similar or contrasting subject.

I'm glad you've found my answers helpful. I'm about to start studying to become a teacher of English as a second language. Writing, which is what I do, doesn't pay enough.

I like answering questions for which the answers are not immediately obvious. And it is always nice when a poster is polite and shows their appreciation for answers received.