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Do equivalents equivalent equivalents? Like equals equal equals?

If I have to show that the given four statements are equivalent, can I just show that the 1st is equivalent to the 2nd, 2nd to 3rd, and 3rd to 4th? Or do I have to show all of them separately?

ryang
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Arunima
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  • That is IF you can prove that “equivalent” is transitive – insipidintegrator Jun 16 '22 at 18:45
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    If you want to show that statements $P,Q,R,S$ are all equivalent, you can do that by way of showing $P\iff Q$ and $Q\iff R$ and $R\iff S$, noting that to show something like $P\iff Q$ requires showing both $P\implies Q$ as well as $Q\implies P$. In doing so, you do get to skip having to individually show $P\iff R$ since if we had $P\iff Q$ and $Q\iff R$ those together to imply that $P\iff R$ must also hold. That all being said, you can do this faster perhaps by instead showing that $P\implies Q$, $Q\implies R$, $R\implies S$ and finally $S\implies P$. – JMoravitz Jun 16 '22 at 18:46
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    Yes, you can prove that $N$ statements $A_1, A_2, ... A_N$ are equivalent using $N$ proofs, "round-robin style", rather than $N(N-1)$ proofs. If you prove $A_1\implies A_2$, ..., $A_{N-1}\implies A_N$, $A_N\implies A_1$, then you've proved that $A_i\implies A_j$ for all $i, j = 1, ... N$. – BrianO Jun 16 '22 at 23:17
  • Point #4 of this answer shows three different ways (among others) to prove the equivalence of $5$ sentences. – ryang Jun 17 '22 at 14:09

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