If $P$ and $Q$ are statements,
$P \iff Q$
and
The following are equivalent:
$(\text{i}) \ P$
$(\text{ii}) \ Q$
Is there a difference between the two? I ask because formulations of certain theorems (such as Heine-Borel) use the latter, while others use the former. Is it simply out of convention or "etiquette" that one formulation is preferred? Or is there something deeper? Thanks!
$\endgroup$ 13 Answers
$\begingroup$As Brian M. Scott explains, they are logically equivalent.
However, in principle, the expression $$(*) \qquad A \Leftrightarrow B \Leftrightarrow C$$ is ambiguous. It could mean either of the following.
$(A \Leftrightarrow B) \wedge (B \Leftrightarrow C)$
$(A \Leftrightarrow B) \Leftrightarrow C$
These are not equivalent; in particular, (1) means that each of $A,B$ and $C$ have the same truthvalue, whereas (2) means that either precisely $1$ of them is true, or else all $3$ of them are true. Also, you can check for yourself that, perhaps surprisingly, the $\Leftrightarrow$ operation actually associative! That is, the following are equivalent:
- $(A \Leftrightarrow B) \Leftrightarrow C$
- $A \Leftrightarrow (B \Leftrightarrow C)$.
In practice, however, (1) is almost always the intended meaning.
$\endgroup$ 13 $\begingroup$They are exactly equivalent. There may be a pragmatic difference in their use: when $P$ and $Q$ are relatively long or complex statements, the second formulation is probably easier to read.
$\endgroup$ 1 $\begingroup$"TFAE" is appropriate when one is listing optional replacements for some theory. For example, you could list dozen replacements for the statements, such as replacements for the fifth postulate in euclidean geometry.
"IFF" is one of the implications of "TFAE", although it as $P \rightarrow Q \rightarrow R \rightarrow P $, which equates to an iff relation.
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