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Let us assume the following definition:

`` S is said to be A if S satisfies the condition C. '' -----------(P)

Can it mean that:

`` S is A if and only if S satisfies the condition C. '' ---------- (Q)

From the statement (P), I have the following argument: If S satisfies condition C, then S is A. Conversely, if S is A, it means S has already satisfied the condition C, so the converse also holds. And thus, the statement (P) means the statement (Q).

Many times I have seen that something is defined in some book, and the same definition has been written as a theorem/result containing the phrase `if and only if'.

My query is: Does the statement (P) mean the statement (Q)?

Kevin.S
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    Usually a definition is written with a "single if", and not iff. So for example a number is rational if there exists... Note I could have written iff here. – Eminem Jan 17 '21 at 11:01
  • @Eminem I don't now if this is a cultural convention, but in Italy we use iff. – Kandinskij Jan 17 '21 at 11:11
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    All definitions are "if and only if" statements, but they are usually written as "if" statements, the "only if" being understood. – Gerry Myerson Jan 17 '21 at 11:45
  • While if and only if is always meant, I actually like using if. By using if you implicitly leave the door open to generalising your concept. For example, by not saying a continuous function is a function from $\mathbb{R}^n\to\mathbb{R}^m$ satisfying ..., you allow for later definitions of continuity in arbitrary topological spaces. – Thomas Anton Jan 17 '21 at 12:59
  • @Thomas taken literally, that causes a problem the moment someone tries/is asked to prove that something is not continuous. – Mark S. Jan 17 '21 at 13:16
  • @MarkS. Absolutely. But I just find it to be a nice allusion, not a literal statement. – Thomas Anton Jan 17 '21 at 13:24

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