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In the following two sentences, I think that the first one is more precisely correct:

(1) Let G be a graph with n vertices, and S be the set of all subgraphs of G.

(2) Let G be a graph with n vertices, and let S be the set of all subgraphs of G.

gete
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    They're both precisely correct according to the rules of grammar. Merely variants on a theme. (1) displays an example of the appliance of the conjunction reduction rule. Would you say "I am hot and I am weary" is 'more correct' than "I am hot and weary"? // Of course, there are occasions where deletions of any kind aree better avoided as they give rise to ambiguity or difficulty in parsing. – Edwin Ashworth Nov 26 '20 at 16:58
  • They are both grammatical, but the more concise (1) would perhaps be preferred by most speakers. "Let G [be a graph with n vertices], [and __ S [ be the set of all subgraphs of G]]. Syntactically, the be clauses form a coordination, each serving as second complement of "let" or gap __. – BillJ Nov 26 '20 at 18:31
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    You may even use a further ellipsis and not repeat "be". – LPH Nov 26 '20 at 18:54
  • Thanks to everyone for the comments.. – gete Nov 27 '20 at 02:48

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