Is Commutativity of Addition True for Integers?

[iwonde]

Homework Statement

Prove that for any two integers a and b, a+b=b+a. You may use the face that this holds for natural numbers.

Homework Equations

The Attempt at a Solution

a=(x,y), b=(u,v)
x,y,u,v are natural numbers
a+b = (x,y)+(u,v) = (u,v)+ (x,y) = b+a

I'm not sure if my attempt was correct.

 

[Billy Bob]

(x,y)+(u,v) = (u,v)+ (x,y)

This step assumes the conclusion. You have to prove it.

Hint: What is the definition of (x,y)+(u,v)?

 

[Mark44]

Homework Statement

Prove that for any two integers a and b, a+b=b+a. You may use the face that this holds for natural numbers.

Homework Equations

The Attempt at a Solution

a=(x,y), b=(u,v)

What do the equations above mean? a and b are integers, not ordered pairs.

x,y,u,v are natural numbers
a+b = (x,y)+(u,v) = (u,v)+ (x,y) = b+a

I'm not sure if my attempt was correct.

 

[Hurkyl]

What do the equations above mean? a and b are integers, not ordered pairs.

I suspect the opening poster forgot to tell us that he's working with a specific presentation of the integers -- represented as the set of ordered pairs of natural numbers modulo an equivalence relation -- along with the definition of + that he's using.

(Of course, maybe that's exactly what you were prompting the opening poster to say -- if so, sorry 'bout that)

 

[Mark44]

Nope, I wasn't thinking that at all, so no problem.