# Proving 0v=0 using only the 10 Axioms

• torquerotates
In summary, the conversation discusses the use of axioms to prove mathematical theorems and the importance of following logical reasoning. It is mentioned that in order to prove a statement using axioms, no assumptions should be made except for the given hypothesis. It is also mentioned that the use of definitions and binary operations does not necessarily require additional axioms to be proven.
torquerotates
Well, I'm supposed to prove 0v=0

It is stated that I'm only allowed to use the following axioms.

let a,b,c be vectors and V is a vector space, then
1)a&b is in V then a+b is in V
2)a+b=b+a
3)a+(b+c)=(a+b)+c
4)0+a=a+0=a
5)a+(-a)=(-a)+a=0
6)a is in V implies ka is in V
7)k(a+b)=ka+kb
8)(k+m)a=ka+ma
9)k(ma)=(km)a
10) 1a=a

The book does it like this, and i think its wrong

0v=(0+0)v=0v +0v { axioms 4&8}

now subtract 0v from both sides { axioms ?}
we get 0=0v

you see the problem here? there's no justification for the subtraction step because there is no axiom allowing the step. Logically I assume that I'm only allowed to use the 10 axioms to prove this theorem.

Actually I figured it out

0v=(0+0)v=0v+0v

Now I just use another equation that has nothing to do with the first.
namely, 0v+(-0v)=0
well, this implies 0v+(-0v)=0=(0v+0v)+(-0v)
implies 0=0v+(0v+-0v)=0v+0=0v

But I was working on another problem in which the author uses incorrect reasoning.

authors' proof:
Prove: if a+c=b+c, then a=b
(a+c)+(-c)=(b+c)+(-c) add (-c) to both sides
a+(c-c)=b+(c-c)
a+0=b+0
a=b

you see how in this example, (-c) was actually added to both sides. There is no way to justify this according to the 10 axioms.

you don't need a axiom to add an element to both sides of a equallity, that is how it works.

That is if

a = b then a+c=b+c

this is not a axiom, this is how equality works, he proves that

a = b if and only if a+c=b+c

You will do a lot better in math if you stop assuming anytime you don't understand something, the author is wrong.

You don't have to have an axiom that says "if a= b then a+ c= b+ c", that's part of the definition of "binary operation" and is assumed whenever you have a binary operation.

Its not a+c=b+c iff a=b. Its a+c=b+c implies a=b. If it is just a binary operation,(I presume you guys to mean it is a definition), then why can it be proved using axioms only? No assumptions are needed except the hypothesis. Having the need to create a definition merely indicates that the statement cannot be proven.proof: (a+c)+(-(a+c))=0 ax.5
(a+c)+(-(b+c))=0 hypothesis
(a+c)+((-b)+(-c)))=0 ax.7
(a+c)+((-c)+(-b))=0 ax2
((a+c)+(-c))+(-b)=0 ax3
(a+(c+(-c)))+(-b)=0 ax3
(a+0)+ (-b) =0 ax5
(a)+(-b)=0 ax.4
a=b ax.5

(a)+(-b)=0 ax.4

to

a=b ax.5

what do you use there?

yep, you are right you use what you are trying to prove (a+c=b+c implies a=b), just with
a -> a+(-b)
b-> 0
c-> b

so you haven't proved anything

ax. 5 says

a+(-a)=(-a)+a=0

not

a+(-b)= 0 => b=a

a=a+0 ax4
a=a+(c+(-c)) ax5
a=(a+c)+(-c) ax3
a=(b+c)+(-c) hypothesis
a=b+(c+(-c)) ax3
a=b+0 ax5
a=b ax4

Well? Haven't I disproven that the binary operation is "merely" a definition?

Well, in the defense of the last step my first proof,
a+(-b)=0

b+0=b=(a+(-b))+(b)=a+(b+(-b))=a+0=a ax.4,3,5&4 in order form left to right.
therefore a=b

## What does it mean to prove 0v=0 using only the 10 Axioms?

Proving 0v=0 using only the 10 Axioms means to logically and mathematically demonstrate that the product of any number and 0 is equal to 0, using only the 10 Axioms of Algebra.

## Why is it important to prove 0v=0 using only the 10 Axioms?

Proving 0v=0 using only the 10 Axioms is important because it serves as a fundamental concept in algebra and lays the foundation for more complex mathematical concepts. It also helps to develop critical thinking and problem-solving skills.

## What are the 10 Axioms used to prove 0v=0?

The 10 Axioms used to prove 0v=0 are the Axioms of Closure, Commutativity, Associativity, Distributivity, Identity, Inverses, Zero, Multiplicative Identity, Multiplicative Inverses, and Transitivity.

## Can 0v=0 be proven using other methods besides the 10 Axioms?

Yes, 0v=0 can also be proven using other mathematical principles and theorems, such as the properties of multiplication and the concept of equality. However, using the 10 Axioms provides a more direct and concise proof.

## How can proving 0v=0 using only the 10 Axioms be applied in real-life situations?

The concept of 0v=0 is applicable in various fields, such as engineering, physics, and economics. For example, in engineering, it can be used to calculate the total force on an object with a mass of 0, and in economics, it can be used to understand the effects of multiplying a quantity by 0 on a given situation.

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