I often see "The following are equivalent" in theorems that proceed to only have two statements, (a) and (b) be equivalent to each other. When is this appropriate to do rather than say "if and only if"?

  • $\begingroup$ I struggled to find a tag for this since it's not subject area specific. $\endgroup$ – Irregular User Jan 23 '16 at 19:55

I can think of a couple cases:

  • The sentence would be too long or complicated to parse if you wrote "[condition] if and only if [condition]"
  • You want to explicitly label the two conditions (a) and (b) so you can reference them in the proof.
  • 1
    $\begingroup$ +1. Considerations of layout and/or convenience within the proof or within a later proof would make one prefer this presentation. It looks and reads better to refer to "(b) of Lemma 3" than to say "the righthand side of Lemma 3" or "RHS of Lemma 3". All that said, yes sometimes it looks a little funny to read "the following are equivalent" only to find that what follows are just two propositions. $\endgroup$ – BrianO Jan 23 '16 at 23:32

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