Prove 0 + 0 = 0 (with properties and field axioms)

[jtayag0622]

Hi,
I encountered this weird problem. the instruction is to prove 0 + 0 = 0. I have to justify my steps with properties and field axioms.
I tried to do it like this:
x = x Reflexive Property
x + -x = -x + x Operational Property of Equality (OPE)
x + -x + 0 = -x + x identity
0 + 0 = 0 Inverse property of Addition

Please help. I don't if that's even correct. Thanks!

 

[JeffM]

I am not going to attempt to provide a proof without knowing just what axioms and theorems you can use, but I do have a question.

From line 2 to 3, you go

\(\displaystyle x + (-\ x) = (-\ x) + x \implies \{x + (-\ x)\} + 0 = (-\ x) + x.\)

That strikes me as valid only if you have an axiom or theorem of the form

\(\displaystyle a + 0 = a \text { for any } a.\)

But if you do, then

\(\displaystyle a + 0 = a \text { for any } a \implies 0 + 0 = 0.\)

What am I missing?

 

[pka]

Hi,
I encountered this weird problem. the instruction is to prove 0 + 0 = 0. I have to justify my steps with properties and field axioms.
I tried to do it like this:
x = x Reflexive Property
x + -x = -x + x Operational Property of Equality (OPE)
x + -x + 0 = -x + x identity
0 + 0 = 0 Inverse property of Addition
Please help. I don't if that's even correct. Thanks!

In any list of the field axioms one of which states: \(\displaystyle (\exists 0\in\mathcal{F})(\forall a\in\mathcal{F})[0+a=a]\)
Letting \(\displaystyle a=0\) gets the job done.

 

[sticky-tiles]

Interesting

In any list of the field axioms one of which states: \(\displaystyle (\exists 0\in\mathcal{F})(\forall a\in\mathcal{F})[0+a=a]\)
Letting \(\displaystyle a=0\) gets the job done.

so interesting, 0 means nothing

 

[Steven G]

so interesting, 0 means nothing

What does 0 means nothing mean?? Does it not exist? If it means nothing then it can't mean the additive identity. Then what would a + (-a) equal? A number would no longer have an additive inverse if 0 meant nothing.

List of things about 0
1) additive identity
2) allows for additive inverses
3) You can't divide by 0
4) 0 times any valid number is 0
5) neither positive nor negative
6) ) only has one number on it's multiplication table
...

Can you tell me things about 2 or 3 or 4 other than their times table and a few others.

0 certainly does not mean nothing.

 

[Dr.Peterson]

so interesting, 0 means nothing

In English, this is an ambiguous statement, which therefore shouldn't be made, as it confuses people. Rather, mathematically, zero is the size of nothing: that is, the number of elements in an empty set. If zero were literally nothing, then it couldn't be used in arithmetic!

Addition (of natural numbers) means finding the size of a set formed by combining two disjoint sets (that is, two sets that have no elements in common). For example, adding 2 + 3 finds the number of elements in the union of the sets {1, 2} and {a, b, c}: the set {1, 2, a, b, c} contains 5 elements.

So adding 0 means adding no elements to a set: the sum 3 + 0 is the number of elements in the set {1, 2, 3} combined with the set {}. Clearly, the result is 3.

And adding 0 + 0 means finding the number of elements in the set {}U{} = {}, so it is zero.

This is how we might prove the requested fact from a definition of addition of whole (or natural) numbers. But, as has been pointed out, the original question assumed knowledge of the "field axioms", which are facts that will already have been proved about some set (such as the rational numbers or the real numbers -- they are not all true of the natural numbers) in order to call that set a "field". These axioms are listed here and here. One of them is "a + 0 = a", for any element a. Since 0 is an element of the field, this immediately implies that 0 + 0 = 0.