## Verification of Simple Proof

1. The problem statement, all variables and given/known data

I am using Spivak's Calculus and just finished the third exercise in part 1.
It was a very easy exercise but it seems that Spivak makes some assumptions.
The problem is as stated:

If x$^{2}$=y$^{2}$, then either x=y or x=-y. Prove it.

The proof was relatively simple (by factoring out (x-y)(x+y) from x$^{2}$
-y$^{2}$ and showing that either (x+y)=0 or (x-y)=0 or both). What I had problems with was that it was assumed that a(0)=0. So, I will now propose a proof that a(0)=0 for all real a using the properties listed in the book. All I need is a confirmation that my proof haw no flaw. Thank you.
2. Relevant equations

3. The attempt at a solution

Introduce (a*0).

(a*0)+(a*0)=0+(a*0).

By the distributive law

a(0+0)=(a*0)+0

By the existence of the additive identity

a*0=(a*0)+0.

Subtracting (a*0) from both sides yields

(a*0)-(a*0)=0

And by the existence of additive identity

0=0

Which is true. Is this valid?
 PhysOrg.com science news on PhysOrg.com >> Ants and carnivorous plants conspire for mutualistic feeding>> Forecast for Titan: Wild weather could be ahead>> Researchers stitch defects into the world's thinnest semiconductor

Recognitions:
Gold Member
Staff Emeritus
 Quote by kripkrip420 1. The problem statement, all variables and given/known data I am using Spivak's Calculus and just finished the third exercise in part 1. It was a very easy exercise but it seems that Spivak makes some assumptions. The problem is as stated: If x$^{2}$=y$^{2}$, then either x=y or x=-y. Prove it. The proof was relatively simple (by factoring out (x-y)(x+y) from x$^{2}$ -y$^{2}$ and showing that either (x+y)=0 or (x-y)=0 or both). What I had problems with was that it was assumed that a(0)=0. So, I will now propose a proof that a(0)=0 for all real a using the properties listed in the book. All I need is a confirmation that my proof haw no flaw. Thank you. 2. Relevant equations 3. The attempt at a solution Introduce (a*0). (a*0)+(a*0)=0+(a*0).
"Introduce (a*0)" in what??.

 By the distributive law a(0+0)=(a*0)
That's not the distributive law- that's the fact that 0+ 0= 0. And from that it follows that a(0)+ a(0)= a(0)

 By the existence of the additive identity a=a*0
That is certainly NOT true! And I don't see any point in talking about the "existence of the additive identity" when you have already been using "0" which is defined as the additive identity.

 And we have proven our proposition. Is this valid?
No, it is not. Instead, from the distributive law a(0+ 0)= a(0)+ a(0). And since 0 is additive identity, x+ 0= x for any x. In particular 0+ 0= 0 so the left side is just a(0). Once you have a(0)= a(0)+ a(0) use the fact that a(0) has an additive inverse.

Recognitions:
Gold Member
Staff Emeritus

## Verification of Simple Proof

Same errors I noted before. You seem to have payed no attention to my first post.
 The easiest way to do it would probably be just to give it some modus tollens imo I'm not quite sure what you're trying to do by playing about with 0 here...
 Okay. I will redo it here and explain myself better. In Spivak's book, he lists 12 properties of the real numbers. I will use those properties to prove that for any real a (a*0)=0. Begin with the assumption that the above statement is false. Since (a*0) is in R, then by the closure under addition property, (a*0)+(a*0) must be in R. Therefore, we can add (a*0) to both sides of the above equation and we shall have (a*0)+(a*0)=0+(a*0). (1) Now, the left side of the equation can be factored out by the distributive law to yield (a(0+0))=(a*0)+(a*0). Thus, (1) becomes: (a(0+0))=0+(a*0). (2) But, 0+0=0 and the left side becomes (a(0+0))=(a(0)). Thus, for (2), we have: (a(0))=0+(a(0)). (3) But, by property 3 in Spivak's book, for every real k, there exists it's additive inverse denoted -k such that k+(-k)=0. Therefore, by adding the negative of (a*0) to both sides of equation (3), we get (a(0))+(-(a(0)))=0+(a(0))+(-(a(0))). (4) Again, by Spivak's third stated property, we have: (a(0))+(-(a(0))=0 and equation (4) becomes 0=0+0. (5) But it is clear from Spivak's second property that 0+0=0 and equation (5) becomes 0=0. But, this is obviously true, and since valid manipulation was made to our initial equation (a*0)=0, then our initial assumption was false and, therefore, (a*0)=0 must be true. I really hope this clarifies things.

Blog Entries: 8
Recognitions:
Gold Member