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?
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, 2017 at 20:57
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$\begingroup$ @Raphael So there are no counterexamples? $\endgroup$– TurboDec 12, 2017 at 20:59
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$\begingroup$ 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. $\endgroup$– Wei ZhanDec 12, 2017 at 21:03
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$\begingroup$ @WillardZhan so $\mathcal A^\mathcal C=\mathcal A^\mathcal D$? $\endgroup$– TurboDec 12, 2017 at 21:07
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