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Yes, you are correct. Sentence #1 is missing the word "and" between "mine" and "the other." The correct sentence structure should be:
"There are two books; one is mine, and the other is my friend's."
To determine if a sentence is correct, you can double-check the grammar and punctuation. In this case, adding the conjunction "and" between "mine" and "the other" helps to properly connect the two parts of the sentence.