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### Should you use an equivalence or an implication in a definition?

In a definition like the following: An object $x$ is called $P$ if [and only if] it has properties $p$ and $q.$ should one use an implication (if) or an equivalence (if and only if)? It makes ...
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### What is the difference between an axiomatization and a definition?

It is sometimes said that some things are not ever defined but instead axiomatized. What does this mean exactly and what is the difference? For example in set theory the symbol $\in$ is not ever ...
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### Why not allow creativity of definitions?

It appears to me that a fair number of issues with allowing ZFC to work with other mathematical topics is that one cannot phrase certain definitions inside ZFC. Would not this be fixed by allowing ...
58 views

### Having trouble forming mathematical definition

Now I am a little bit confused about the mathematical definitions: Is that all the mathematical definition must be written in form like $(A \stackrel{def}{\equiv} B)$? Can they be written in form like ...
149 views

### Definition of “definition”: use iff or if? [duplicate]

There are topics with the same name but my question is not as abstract as in those. My question is as follows: taken a generic definition like $x\;\mathbf{ is\; something}$ if $y$ it could be written ...
28 views

### Which one is a correct way to give definitions? [duplicate]

When reading books I pay attention that some of them giving a definition to some notion use "if" but others "if and only if". For example, A set is called empty if it has no elements A set is called ...
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### If then statement regrading definition of lognormal distribution and the inverse of that statement?

Hello I'm learning about probability distributions and this was the given definition for a lognormal distribution: A random variable X follows a lognormal distribution if its natural logarithm, ln ...
117 views

### First order logic “abbreviation”

I am curious what justifies the use of shorthands or abbreviations for certain formulas in first order logic. In particular, I'm interested in building up some basic mathematical principles from the ...
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### Confusing definition about a free monoid and comparison operator

In one of the works about tree languages I met the following definition: Let $\mathcal{N}$ be the set of nonnegative integers and let $U$ be the free monoid generated by $\mathcal{N}$ with ...