# On query equally powerful oracles?

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?

• If a=b then f(a) = f(b), provided f is a well-defined function. I don't understand what the question here is. – Raphael Dec 12 '17 at 20:57
• @Raphael So there are no counterexamples? – Bread Winner Dec 12 '17 at 20:59
• 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. – Willard Zhan Dec 12 '17 at 21:03
• @WillardZhan so $\mathcal A^\mathcal C=\mathcal A^\mathcal D$? – Bread Winner Dec 12 '17 at 21:07