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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 ...
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Bi-conditionals

I cannot understand the meaning of if-and-only-if in definitions. For example, when we define a planar graph, we state that "A graph is G is planar, if-and-only-if it has no crossing edges". What does ...
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If a set is open, does it mean that every point is an interior point? [duplicate]

In Walter Rudin's Principles of Mathematical Analysis he defines open set as: "E is open if every point of E is an interior point of E." So this can be translated in logic as "If every point of E is ...
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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 ...