Two sentences:

1) The race among the three relay teams was close; none of the three; however, is giving up hope of winning.
2) The race among the three relay teams was close; however, none of the three is giving up hope of winning.
Which is the correct one grammatically? According to our book, the first one it, but in my opinion, the second one is. Why am I wrong? (or why am I right?)

1 is incorrect because there should not be a semi colon after "three."

The correct sentence, grammatically speaking, is the second one: "The race among the three relay teams was close; however, none of the three is giving up hope of winning."

The reason why your book may have marked the first sentence as correct is due to an outdated grammar rule called the "rule of proximity" or "rule of placement." This rule suggests that the relative pronoun or phrase should be placed as close as possible to the antecedent it refers to.

In this case, the relative pronoun "however" is meant to modify the entire previous clause "The race among the three relay teams was close," implying that despite the closeness of the race, all three teams are still hopeful of winning. According to the rule of proximity, "however" should be placed immediately after the clause it modifies. Therefore, the first sentence might seem correct according to this rule, as it places "however" right after the clause.

However, it is important to note that this rule is more of a style preference rather than a strict grammatical rule. Modern grammar usage allows for more flexibility in sentence structure. In the case of the second sentence, the use of "however" after the semicolon still conveys the intended meaning clearly, even if it does not adhere to the rule of proximity. This sentence is considered more natural and widely accepted in modern usage.

So, while your personal preference for the second sentence is valid, the book may have marked the first sentence as correct due to the outdated rule of proximity.