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### Alternative ways to say “if and only if”?

There are some scenarios about which I would like to get some confirmation: when defining a concept A, We call A, if ... [definition of concept A] Does "if" here mean equivalence instead of ...
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### The set of functions that are zero almost everywhere is enumerable

I have become somewhat overwhelmed with a problem I am working on I had a friend tell me that my proof was wrong. I would be grateful if someone could explain why I am wrong, and possibly offer a ...
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### “Only if” in the set of proposition definition

In my mathematical logic book, the language of propositional logic and the set of well formed formulas are defined with the following definitions: Language of propositional logic The language of ...
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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 ...
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### Conditionals in definition of Strictly Increasing Function

I have a question concerning the definition of strictly increasing function, that I cannot really figure out. The definition reads: Definition: A function $f : \mathbb{R} \to \mathbb{R}$ is ...
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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 ...
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 ...
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 ...