A sentence (or a part of the sentence) is underlined. Four alternatives are given to the underlined part which will improve the sentence. Choose the correct alternative. In case no improvement is needed, select "No improvement". Sentence: "Both of them have not apologized yet."

Introduction / Context:
This question tests your understanding of negative structures and subject–verb agreement when referring to two people. The original sentence "Both of them have not apologized yet" is grammatically understandable but stylistically awkward and not the best standard English. The exam expects you to replace it with a more natural and correct structure expressing that none of the two people has apologised so far.

Given Data / Assumptions:

• Original sentence: "Both of them have not apologized yet."
• We want to express that zero out of two persons has apologised.
• Options: Neither of them has, Any of them have, None of the two, No improvement.
• The subject refers to two people only ("them").

Concept / Approach:
In standard English, when we want to say that not one of two people has done something, we usually use "neither of them" followed by a singular verb: "Neither of them has apologized yet." The phrase "both of them have not" is a clumsy negative construction and can be misinterpreted. The options "any of them" and "none of the two" are also problematic or less idiomatic. The correct approach is to use "neither" for two items and to ensure singular verb agreement.

Step-by-Step Solution:
Step 1: Understand the intended meaning: zero out of two people have apologised up to now. Step 2: Examine option (a): "Neither of them has". This correctly means not one of the two has done the action and takes the singular verb "has". Step 3: Examine option (b): "Any of them have". "Any" is usually used with more than two items and is not the usual choice when referring specifically to two people. Also, the verb "have" would not match the singular phrase "any of them" in formal usage. Step 4: Examine option (c): "None of the two". Although languages sometimes allow "none of the two", the more idiomatic choice for two people is "neither of the two". Also, the phrase here is incomplete and does not include the verb. Step 5: Examine option (d): "No improvement". Keeping "Both of them have not apologized yet" is grammatically clumsy and less natural. Step 6: Conclude that "Neither of them has" is the best replacement for the underlined part.

Verification / Alternative check:
Rewrite the full sentence with the chosen improvement: "Neither of them has apologized yet." This is a standard sentence structure used by native speakers and in grammar books. It clearly indicates that not one of the two individuals has offered an apology up to this point. The verb "has" correctly agrees with the singular idea of "neither". In comparison, "Both of them have not apologized yet" sounds awkward and is rarely used in careful written English.

Why Other Options Are Wrong:

• Any of them have: "Any" is typically used when there are three or more options; also, the combination of "any of them have" is less natural when referring specifically to two people.
• None of the two: While "none of the two" might sometimes be understood, it is not the preferred or most idiomatic expression; "neither of the two" or simply "neither of them" is better.
• No improvement: The original wording is not the best form; standard English prefers the more concise and clear "neither of them has".

Common Pitfalls:
Students often struggle with negative constructions involving "both" and "neither". Remember that "both" emphasises inclusion ("both of them have apologized") while "neither" emphasises exclusion ("neither of them has apologized"). Using "both ... not" can lead to confusion or awkward phrasing. To avoid mistakes, prefer "neither of them" for a clear negative statement about two people.

Final Answer:
The best improvement is "Neither of them has", giving the complete sentence: "Neither of them has apologized yet."