0

I know that to prove $A$ is equivalent to $B$, I have to assume $A$ then prove $B$ and then assume $B$ and prove $A$. But let’s say that $C$ is a axiom. Can I use $C$ in my proof of $B$ from $A$ and vice versa? Or can I only use $A$ and $B$, and not $C$?

Edit: I can only prove $(A \land C) \equiv (B\land C)$. Is there any way to prove $A \equiv B$? Remember $C$ is an axiom.

phst
  • 586
  • I think I’m not clear enough. I can only prove $(A \land C) \equiv (B \land C$). Does that mean $A \equiv B$? – phst Feb 06 '20 at 18:14
  • 2
    If $C$ is an axiom, it is taken as true, so it can be used for any proof. – Paul Feb 06 '20 at 18:20
  • Can you show your proof that $(A\land C)\equiv (B\land C)$? You said you can prove that. How so? – amWhy Feb 06 '20 at 18:34
  • $A$ and $B$ are just placeholders for longer statements. – phst Feb 06 '20 at 18:35
  • You can start with $A, C$, to try to prove $B$. But that gets you only $A\land C$. If you start with $B, C$ to try to prove A, that gets you only $B \land C$. How do you prove therefore $A\land C \equiv B\land C$, unless that's given? I am sorry, but there is not enough information in your post. – amWhy Feb 06 '20 at 18:37
  • If A is a tautology, and B a contradiction, while C is an axiom, then we will never have $A\land C \equiv B\land C$. However, regardless of the truth values of $A, B$, if C is an axiom, then $A \lor C \equiv B\lor C$. Do you see the difference? – amWhy Feb 06 '20 at 18:50
  • $A$ and $B$ are definitely not tautologies, or contradictions. I managed to prove $A \land C \rightarrow B$. This implies $A \land C \rightarrow B \land C$. I also managed to prove $B \land C \rightarrow A$, and thus $B \land C \rightarrow A \land C$. So $A \land C \equiv B \land C$. – phst Feb 07 '20 at 07:46
  • Anyways, I realised that $\bigl( (A \land C) \equiv (B \land C) \bigr) \land C \rightarrow (A \equiv B)$. Since $C$ is an axiom, and I proved $(A \land C) \equiv (B \land C)$, therefore $A \equiv B$ is true. – phst Feb 07 '20 at 07:49

0 Answers0