Terminology – Alternative Ways to Say ‘If and Only If’

Question

There are some scenarios about which I would like to get some confirmation:

1. when defining a concept A,

   We call A, if … [definition of
   concept A]

   Does "if" here mean equivalence
   instead of just sufficiency? Is it
   incorrect to replace "if" with "if
   and only if"?

2. For precisely what condition is
   B satisfied?

   Does "precisely" mean asking for
   necessary and sufficient condition
   for B?

3. Some other ways to say "if and only
   if" / "necessary and sufficient"?

Some references that summarize some standard terminologies in Mathematics such as this one?

Thanks and regards!

Answer 1

1. In definitions, it is very common practice to use "if" even though "if and only if" is meant. (Personally, I always use "if and only if" explicitly). So you would not have trouble finding a book that said something like

   Definition. A group $G$ is simple if $G$ is nontrivial and whenever $N\triangleleft G$, either $N=\{1\}$ or $N=G$.

   But such a definition is meant to be understood to be saying that $G$ is simple if and only if the condition is met.

2. "Precisely" is asking for a condition that is both necessary and sufficient.

3. "Exactly when"; "if ... then ... and conversely", among others.