Linked Questions

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
493 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
237 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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