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Is the following line true?

Consider three problems A, B and C. If $A$ $<p$ $B$ and $B$ $<p$ $C$ and $B$ is NP-complete problem, then $C$ is also NP-complete.

If B is NP-complete, then C would have also been NP-complete if it was in NP but it is not given here. Can we consider it false? Or is there any other reason?

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    $\begingroup$ You are correct, all we know about $C$ is that it is $\mathsf{NP}$-hard. By the way, the fact that $A \leq^m_p B$ is irrelevant. $\endgroup$ Jun 13, 2021 at 9:08

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