Linked Questions

6
votes
11answers
2k views

Why every definition is an “iff”-type statement? [duplicate]

Suppose that we are trying to define a mathematical object $M$. The statement of the definition generally takes the form (or some of its equivalent variant), A mathematical object is said to be $M$ ...
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
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. ...
-1
votes
2answers
198 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 ...
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
111 views

Are math definitions iff statements? [duplicate]

I was wondering if definitions in mathematics are "if and only" statements? I know for sure that theorems are not "iff" statements. Thank you in advance for your help.
1
vote
2answers
200 views

Why do we use “if” in the definitions instead of “if and only if”? [duplicate]

I often write my notes as logical statements and constantly wonder why people use only the "if" direction in the definitions instead of the "if and only if". Consider: "A homomorphism $\phi$ is said ...
0
votes
0answers
80 views

Why are definitions written as 'if-then' statements instead of 'if-and-only-if' [duplicate]

An example from Rudin would be: (c) if $x + y = 0$ then $y = -x$. There may be times when one would have to use the fact that since $y = -x, x + y = 0$. While this is fairly intuitive, professors ...
0
votes
0answers
59 views

Do axioms say “iff”? (or, is it “if”, or, “only if”). [duplicate]

My guess is that they say "iff", but I wanted to make sure as math books have never made it explicit. To take the field axioms for example, is it fair to say that it says both: $F_x\rightarrow A_x$...
0
votes
0answers
44 views

Use of the word “if” in mathematical definitions [duplicate]

I'm looking at the following definition The random variables $X_{1}, \ldots, X_{d}$ are said to be comonotonic if they admit as copula the Frechet upper bound. I am however not quite sure how to ...
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. ...
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
0answers
34 views

I want clear a point about definitions. [duplicate]

I want know wether "defnitions" are if and only if. For example if a set satsfies all four group axioms we say it is group but then we go other way also. thank you.
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 ...
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\...
20
votes
6answers
4k views

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 ...
6
votes
5answers
778 views

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 ...
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 ...
3
votes
1answer
368 views

defining inequality of natural numbers by case-analysis

If I add to Peano Arithmetic a relation (predicate?) symbol $\leq$ and an axiom $\forall n\forall m(n\leq m \leftrightarrow n=m \lor S(n)\leq m)$, can I prove $\forall n\forall m(n\leq m \to n\leq S(m)...
1
vote
3answers
107 views

Shouldn't syntax definitions make use of “iff” rather than “if”?

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 $\mathcal{L}$...
2
votes
3answers
135 views

Understanding iff [duplicate]

I'm having difficulty understanding why it is appropriate to use if and only if, something I thought I had a firm grasp on. From Lara Alcock's book, How to Study as a Mathematics Major: ...
1
vote
1answer
143 views

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 ...
2
votes
2answers
175 views

I there a rigorous, mathematical, approach to definitions (denotations)?

In mathematical logic, a definition is treated as an abbreviation - a denotation which simplifies the discourse making it shorter. This is so much so that in a formal theory or a logic we can do ...
1
vote
2answers
236 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
191 views

Implication or Bidirectional in “x is a Prime”

I have a question regarding First Order Logic. I have to express the property "x is a Prime" in First Order logic. So far I have the following solution: $\forall x\;Prime(x) \leftrightarrow \neg \...
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 ...
1
vote
2answers
91 views

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 ...
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 ...
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
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 ...

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