# To prove that a finite set of statements are logically equivalent, how does one proceed?

Let $S = \{s_1,\cdots,s_n\}$ denote a finite set of statements. How might one go about showing that for all $x,y \in S$ it holds that $x \Leftrightarrow y$?

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However is most convenient, given the specific statements $s_1,\dots,s_n$; there is no general rule. If you’re fortunate, it may be possible to order the statements so that you can without too much pain prove that $s_1\to s_2\to\ldots\to s_n\to s_1$, so that you end up proving just $n$ implications, the minimum possible. Somestimes, however, it’s much easier to prove that each of $s_2,\dots,s_n$ is equivalent to $s_1$, so that you end up proving $2(n-1)$ implications. Any set of implications that contains a chain from each $s_i$ to each other $s_k$ is acceptable. –  Brian M. Scott Mar 5 at 2:05