Notation: false equation

[Function]

Hello

 

For a research we have to do in class, I'd like to use mathematically correct notations.

There is, however, one expression from which I don't know how it's written: a false equation.

 

e.g. Fermat's big theorem:

 

[math]\nexists (x,y,z,n) \in \mathbb{N}_0 : x^n+y^n=z^n[/math]

 

[math]\forall(x,y,z,n)\in\mathbb{N}_0 : \left\{x^n+y^n=z^n\right\}=\emptyset[/math]

 

[math]\forall(x,y,z,n)\in\mathbb{N}_0 : \left\{x^n+y^n=z^n\right\}=0[/math]

 

Or just a very simple example:

 

[math]\left\{1=0\right\}=0[/math]

[math]\left\{1=0\right\}=\emptyset[/math]

 

Is any of these notations correct? If not, which one could I use?

 

Thanks.

 

Function

Edited February 28, 2014 by Function

[mathematic]

the = 0 (zero) is wrong. = empty set is correct. However for Fermat's theorem, correct statement includes n > 2.

[John]

There is also the falsum [math]\bot[/math], used to denote a general contradiction. For instance, in your simple example, you might say [math](1 = 0) \vdash \bot[/math], i.e. "1 = 0 yields a contradiction".

 

For the formal statement of Fermat's Last Theorem, there are many ways to go about it, including

 

1. [math](\forall x, y, z \in \mathbb{N})(n \in \mathbb{N}_{>2} \implies x^n + y^n \neq z^n)[/math],

 

2. [math](\forall x, y, z \in \mathbb{N})(\forall n \in \mathbb{N}_{>2})(x^n + y^n \neq z^n)[/math],

 

3. [math](\forall x, y, z, n \in \mathbb{N})(x^n + y^n = z^n \implies n \leq 2)[/math], and

 

4. [math]\{x, y, z, n \in \mathbb{N} \mid (n > 2) \wedge (x^n + y^n = z^n)\} = \varnothing[/math].

Edited March 1, 2014 by John

[Function]

the = 0 (zero) is wrong. = empty set is correct. However for Fermat's theorem, correct statement includes n > 2.

 

Of course. How could I've forgotten that

 

 

There is also the falsum [math]\bot[/math], used to denote a general contradiction. For instance, in your simple example, you might say [math](1 = 0) \vdash \bot[/math], i.e. "1 = 0 yields a contradiction".

 

For the formal statement of Fermat's Last Theorem, there are many ways to go about it, including

 

1. [math](\forall x, y, z \in \mathbb{N})(n \in \mathbb{N}_{>2} \implies x^n + y^n \neq z^n)[/math],

 

2. [math](\forall x, y, z \in \mathbb{N})(\forall n \in \mathbb{N}_{>2})(x^n + y^n \neq z^n)[/math],

 

3. [math](\forall x, y, z, n \in \mathbb{N})(x^n + y^n = z^n \implies n \leq 2)[/math], and

 

4. [math]\{x, y, z, n \in \mathbb{N} \mid (n > 2) \wedge (x^n + y^n = z^n)\} = \varnothing[/math].

 

Thanks! I'll be using the last one, then Seems better to me

[ajb]

...I'd like to use mathematically correct notations.

Just a general comment, there are no correct notations! Notation is notation and it should be consistent and hopefully lend itself to the problems you hope to tackle. Finding the right notation for the problem can be a big help. However, the actual mathematics is independent of the notation and people use different notation all the time. Your real hope is that everyone else can understand what you have written.

[Unity+]

Just a general comment, there are no correct notations! Notation is notation and it should be consistent and hopefully lend itself to the problems you hope to tackle. Finding the right notation for the problem can be a big help. However, the actual mathematics is independent of the notation and people use different notation all the time. Your real hope is that everyone else can understand what you have written.

That depends on the meaning of 'correct.' Usually, there is a guideline for the use of standard notation and it isn't as if a person can create their own notation and get away with it easily. If there wasn't a standard notation then it would make it difficult to keep learning more and more notations of a particular problem or solution.

[studiot]

 

Usually, there is a guideline for the use of standard notation

 

There are almost no 'standard' notations.

 

Many conventions have grown up from convenience.

Some stem from typographical constraints, eg superscript, greek letters and special symbols.

Some from the desire to fit in with a larger scheme eg the use of x, y, z or x₀, x₁, x₂ or x₁, x₂, x₃ for axes,

Some from the simple fact that there are substantially more constants and possible variables in physics and maths than there are in several alphabets put together.

 

Some from convenience in use eg the D operator v Leibnitz dy/dx v Newton's prime notation in calculus.

 

 

And some from the fact that historically different symbols for the same quantity evolved in two different geographical regions or disciplines so your notation depends where and in what subject you are writing.

 

Finally in the teaching on mathematical subjects some consistency is desirable so as students progress through they are not constantly presented with new and different notation for something familiar. Failure to implement this impedes learning and understanding.

Edited March 1, 2014 by studiot

[ajb]

That depends on the meaning of 'correct.' Usually, there is a guideline for the use of standard notation and it isn't as if a person can create their own notation and get away with it easily.

There is not even really standard notation as it can change across different areas of mathematics and indeed author to author. Even if we fixed everything it would not be correct in any meaningful way. Notation is notation and not the mathematics itself.

 

That said, people should avoid inventing their own notation when very common and accepted notation exists in their field.