Linked Questions

45
votes
12answers
8k views

What is exactly the difference between a definition and an axiom?

I am wondering what the difference between a definition and an axiom. Isn't an axiom something what we define to be true? For example, one of the axioms of Peano Arithmetic states that $\forall n:0\...
49
votes
8answers
47k views

Example of Partial Order that's not a Total Order and why?

I'm looking for a simple example of a partial order which is not a total order so that I can grasp the concept and the difference between the two. An explanation of why the example is a partial ...
0
votes
1answer
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 ...
1
vote
1answer
45 views

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 ...
0
votes
2answers
38 views

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. ...
-1
votes
2answers
196 views

$X$ and $Y$ have the same cardinality if and only if there exist a bijection from $X$ to $Y$? [duplicate]

My textbook says "Let $X$ and $Y$ be sets. We say $X$ and $Y$ have the same cardinality if there is a bijection $f: X \to Y$." I was wondering why the text does not say "if and only if." A ...
1
vote
2answers
235 views

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 ...
0
votes
1answer
144 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 ...
0
votes
1answer
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 ...
0
votes
1answer
35 views

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 ...
0
votes
2answers
50 views

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 ...
10
votes
2answers
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 ...
0
votes
1answer
116 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 ...
3
votes
2answers
492 views

“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 ...
0
votes
2answers
156 views

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. ...

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