What is the difference between these two sentences?
This is a book that I brought from the library.
This is a book I brought from the library.
Do both work? If so why do they both work?
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There's no difference between the two sentences; they convey the same meaning.
The word "that" is a relative pronoun in the relative clause "that I brought from the library". You can omit the relative pronoun when it's the object of the clause.
I think the sentence without "that" sounds more common and idiomatic.