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  1. If $\mathcal C=\mathcal D$ then does $\mathcal A^\mathcal C=\mathcal A^\mathcal D$ hold ($\mathcal C^A=\mathcal D^A$ need not hold)? The class $\mathcal A$ could query same for $\mathcal C$ and $\mathcal D$ since $\mathcal C=\mathcal D$ as languages. Correct? Is there a counterexample?

  2. If $\mathcal C\subseteq\mathcal D$ then does $\mathcal A^\mathcal C\subseteq\mathcal A^\mathcal D$ hold ($\mathcal C^A\subseteq\mathcal D^A$ need not hold)? The class $\mathcal A$ could query same for $\mathcal C$ and $\mathcal D$ since $\mathcal C$ is in $\mathcal D$ and simulate $\mathcal A^\mathcal C$ with $\mathcal A^\mathcal D$. Correct?

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  • $\begingroup$ If a=b then f(a) = f(b), provided f is a well-defined function. I don't understand what the question here is. $\endgroup$ – Raphael Dec 12 '17 at 20:57
  • $\begingroup$ @Raphael So there are no counterexamples? $\endgroup$ – Bread Winner Dec 12 '17 at 20:59
  • $\begingroup$ Working as oracles, $\mathcal{C}$ and $\mathcal{D}$ are equal as two sets of languages, instead of 'equal in power' of some models of computation. $\endgroup$ – Willard Zhan Dec 12 '17 at 21:03
  • $\begingroup$ @WillardZhan so $\mathcal A^\mathcal C=\mathcal A^\mathcal D$? $\endgroup$ – Bread Winner Dec 12 '17 at 21:07

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