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Consider the following definition:

A sequence $\{p_n\}$ is Cauchy if we have that for every $n, m \ge N$: $$|p_n - p_m| < \epsilon$$

Although if and only if is not used, we know that if a sequence is Cauchy, the Cauchy criterion holds. Consider now the following definition:

An action of a group $G$ on a set $X$ is transitive if and only if for each $x, y \in X$ there is a $g \in G$ such that $gx = y$.

What motivates the use of "if and only if" here? As far as definitions go, my head has automatically thought "if and only if." The latter example is taken from Ratcliffe's "Foundations of Hyperbolic Manifolds", and he switches a lot between "if" and "if and only if" in his definitions.

Andrew Thompson
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  • http://en.wikipedia.org/wiki/If_and_only_if – Alex Silva Dec 10 '14 at 10:05
  • Yes, I do know what iff means. If you give me an example of a Cauchy sequence where the Cauchy criterion does not hold, I will be happy to delete this question. – Andrew Thompson Dec 10 '14 at 10:06
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    I think definitions are always $\iff$ statements. For instance the definition $A\in M(n\times n)$ is invertible if there exists $B \in M(n\times n)$ such that $AB = BA = E_n$ can also be written, because is a definition, as $A\in M(n\times n)$ is invertible $: \iff$ there exists $B \in M(n\times n)$ such that $AB = BA = E_n$.

    Personally i think the second option is better, because it avoids confusion.

     – Bman72 Dec 10 '14 at 10:06
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    You can also see this post – Mauro ALLEGRANZA Dec 10 '14 at 10:12

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There is no difference between "if" and "if and only if" in a definition. More precisely, in the context of a definition "if" should be read as a shorthand for "if and only if".

I can't comment about Ratcliffe specifically, but my guess is that the reason it appears E is using them interchangeably is because E is.

Eric Stucky
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