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When proving say $P \iff Q$, we generally write the proof in this fashion:

Proof:

$\Rightarrow$ [Proof that $P$ implies $Q$]

$\Leftarrow$ [Proof that $Q$ implies $P$]

QED.

Now, say we wanted to prove $P_1\iff P_2\iff P_3 \iff P_4$. We would show $$P_1\implies P_2\implies P_3\implies P_4\implies P_1$$ But how should the proof look, as it relates to formatting (the arrows and such)?

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Bonnaduck
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    I typically just write $P_1 \implies P_2$, $P_2 \implies P_3$ and so on and so forth, like you have there. Where you have the arrows, I would just indicate what implication you are proving. – A. Thomas Yerger Sep 08 '16 at 01:04
  • While the efficient "round-robin" implications you suggest will accomplish the mutual equivalence of several statements, it is not always necessary to take the statements in pairs just that way. Keep in mind each segment of such a proof should deal with one implication at a time. – hardmath Sep 08 '16 at 01:10

1 Answers1

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Use a similar format, but either numbering or explicitly listing the implications you are proving.

Proof of: $P_1 \iff P_2 \iff P_3 \iff P_4$

$(P_1 \implies P_2)$: Because ... suchansuch reasons

$(P_2 \implies P_3)$: Because ... soandso said so.

$(P_3 \implies P_4)$: Because ... I dunno, it just is.

$(P_4 \implies P_1)$: Because ... ... magic?

Therefore by chaining: $P_1 \iff P_2 \iff P_3 \iff P_4$

$\blacksquare$

But hopefully with better justifications. $\ddot\smile$

Graham Kemp
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