How do you prove it?

[studiot]

It matters because we need to differentiate between the number 'zero', which is a member of quite a few sets, and the situation where there is no member of a set involved in a statement.

 

I have no apples in my bag ie the number of apples in my bag is zero.

 

but

 

If I consider a bag of apples it cannot contain zero apples or it is not a bag of apples.

 

So the set of possible numbers of apples in my bag can contain zero, but the set of possible numbers of apples in a general bag of apples cannot.

 

 

Student or otherwise, let me congratulate you the reasonableness of your discussion style - it will take you far.

Edited April 21, 2013 by studiot

[Amaton]

but

 

If I consider a bag of apples it cannot contain zero apples or it is not a bag of apples.

 

So the set of possible numbers of apples in my bag can contain zero, but the set of possible numbers of apples in a general bag of apples cannot.

 

Sounds reasonable. If we can define [math]0=\left\{\,\right\}[/math], is it not rigorously allowed to represent a "bag of apples" with zero by the empty set?

 

Student or otherwise, let me congratulate you the reasonableness of your discussion style - it will take you far.

 

Haha, I've had more than my share of arrogant statements, but thanks. You also make for well-discussion.

Edited April 24, 2013 by Amaton

[studiot]

Well to continue my bag of apples analogy, would you use the empty set to represent a bag of 5 pears as a set of zero apples?

If so you would be ignoring 5 members of the set, S

How would you perform arithmetic on this set so that for any (a,b) in S (a+b) is in S ?

That would be adding apples to pears, which you could only do if you are considering fruit.

 

There are equivalent situations in number theory and it depends upon what you are doing with your numbers to how you approach them.

 

I call the set you refer to viz - {0,1,2,3,.......} - the set of orninal numbers.

 

I call the set - {1,2,3,...} - the set of natural or counting numbers.

 

You need both of these. The natural numbers allows the Dedekind route to the construction of the real number system, via the integers and rationals.

[Amaton]

There are equivalent situations in number theory and it depends upon what you are doing with your numbers to how you approach them.

 

I call the set you refer to viz - {0,1,2,3,.......} - the set of orninal numbers.

 

I call the set - {1,2,3,...} - the set of natural or counting numbers.

 

You need both of these. The natural numbers allows the Dedekind route to the construction of the real number system, via the integers and rationals.

 

Makes sense. Thanks for the lesson

[studiot]

Apologies, I see I made a spelling mistake the integers, including zero should be the ordinal numbers.

 

For a more mathematical example consider the set of possible numbers of row or columns of a matrix, M. If the matrix is square then M is also the dimension of the (vector) space spanned by the matrix.

 

In any event consider whether the elements of M should be drawn form the ordinals or the natural numbers.

 

If we allow zero what is the effect of multiplying a matrix with another with zero rows?

What does a matrix with zero rows look like?

[LaurieAG]

For a more mathematical example consider the set of possible numbers of row or columns of a matrix, M. If the matrix is square then M is also the dimension of the (vector) space spanned by the matrix.

It's probably easier to consider binary logic, as in discrete computer maths, and the presence of a high current input (1) vs a low current input (0).

 

http://en.wikipedia.org/wiki/Functional_completeness

[studiot]

It's probably easier to consider binary logic, as in discrete computer maths, and the presence of a high current input (1) vs a low current input (0).

 

Please explain.

[Amaton]

If we allow zero what is the effect of multiplying a matrix with another with zero rows?

What does a matrix with zero rows look like?

 

That I cannot answer on my own. I would've suspected it to be undefined. a matrix with both zero rows and columns is an empty matrix, as you may know.

[studiot]

You can't rely on Wikipedia to be 100% accurate, it does not have the benefit of the level of scrutiny enjoyed by conventional textbooks.

 

In the case of that article I think the authors have confused the zero (or null) matrix with a matrix that has no entries.

 

You cannot create a matrix on a computer (or anywhere else) that has no entries. So wht happens is the matrix which has every entry as zero is employed.

 

This also has to happen in matrix operations viz you have to select the entries from a field, which has a zero element, so you can do conventional arithmetic.

 

Remember the rules of matrix operations:

 

Let E be a matrix with no entries, a be a matrix conformable to additions with E and B be a matrix conformable to multiplication with E.

 

Then what matrices do you think are the results of (A+E) = C and EB = D ?

 

C cannot equal A and D cannot equal B since E is neither the additive nor the multiplicative identity.

 

However the rules require that every sum and product can be formed.

 

therefore E is not in the set of all matrices, ie it does not exist.

 

Further, remember that computers start counting from zero, we start counting from 1. ie a byte is 0-255, not 1-256.

[studiot]

Amaton,

You may like to took at my post #73 in this thread, where I show how to construct the ordinals from nothing at all.

 

http://www.scienceforums.net/topic/69384-what-is-nothing/

Edited May 12, 2013 by studiot