Proof of multiple equivalences I want to prove that 5 statements are equivalent. I have been told that it is not necessary to prove $\binom{5}{2} = 20$ equivalences; we can show, for example, that
$P_1 \implies P_2$, $P_2 \implies P_3$, $P_3 \implies P_4$, $P_4\implies P_5$ and $P_5 \implies P_1,$ "completing" the loop.
I've tried something different:
$$P_1 \iff P_5\\
P_2 \iff P_3\\P_1 \land P_2 \land P_3 \land P_5 \iff P_4$$
Is this also correct?
 A: *

*The lack of symmetry in your proposal

*

*$P_1 \iff P_5$

*$P_2 \iff P_3$

*$P_1\land P_2\land P_3\land P_5 \iff P_4$
suggests that they are not jointly equivalent to the original

*

*$P_1 \implies P_2$

*$P_2 \implies P_3$

*$P_3 \implies P_4$

*$P_4\implies P_5$

*$P_5 \implies P_1.$



*Rob has given a counter-interpretation; here's another one $$P_1
:=\;  7=7\\ P_5 :=\;  7=7\\ P_2 :=\;  8=9\\ P_3 :=\;  8=9\\ P_4 :=\;
8=9,$$ and yet another one $$P_2 :=\;  7=7\\ P_3 :=\;  7=7\\ P_1
:=\;  8=9\\ P_4 :=\;  8=9\\ P_5 :=\;  8=9.$$ The $5$ (non-equivalent) statements in each interpretation
jointly satisfies the first (your proposal)—but not the second
(the original)—set of sentences.


*



*Here's an example set of sentences that is jointly
equivalent to the original set:

*

*$P_1\implies P_2\land P_3\land P_4\land P_5$

*$\lnot P_1\implies\lnot P_2\land \lnot P_3\land \lnot P_4\land \lnot P_5.$
(Assume, without loss of generality, that $P_3$ is true; then, by contraposition, so is $P_1;$ consequently, so are the three remaining statements.)
Another that works is Hagen's last suggestion in their second comment under the OP.
A: Assume that $x$ ranges over the integers. Take $P_1$ and $P_5$ both to be $x \ge 0$. Take $P_2$ and $P_3$ both to be $x \le 1$. Finally take $P_4$ to be $x \ge 0 \land x \le 1$. Then $P_1 \iff P_5$, $P_2 \iff P_3$ and $P_1 \land P_2 \land P_3 \land P_5 \iff P_4$ are all valid, but $P_4$ is, strictly stronger than, and hence not equivalent to, any of the other $P_i$. So given your answer to the comment about your notation, your proposal is not a valid way to prove the equivalence of the $P_i$.
