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### 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 ...
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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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### A concern on the definition of compactness in a metric space [duplicate]

Let $(X,d)$ be a metric space. This space is compact if any sequence $x_n \subset X$ has a convergent subsequence. This is how I'm given the definition of a compact metric space and it confuses me. ...
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### 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 ...
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### Natural deduction: swapping equivalent formulas or definitions

In a natural deduction systems, I sometimes see what are called rules of replacement (also called rules of equivalence). These include equivalences like DeMorgan's Laws, or contraposition. Take the ...
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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 ...
1k views

### 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 ...
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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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### Logic Definitions/Axioms (Are they iff statements?) [duplicate]

Are all definitions or axioms in logic biconditional (iff) statements? It would make sense to me that they would be. A lot of times I will read a definition though and it won't be written as an iff. ...
29 views

### 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 ...
Definitions for the syntax of formal languages frequently make use of clauses such as If $t_1, ..., t_n$ are terms in $\mathcal{L}$ and $P$ is an $n$-ary predicate in the vocabulary of $\mathcal{L}$...
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 ...