Spivak's Calc, Basic Properties of Numbers, Incredibly Discouraged

[katia11]

Homework Statement

Prove: If x²=y² then x=y or x=-y

Homework Equations

the twelve basic properties?

The Attempt at a Solution

I am in an honors calculus course that teaches calculus through proofs. Other than triangle geometry proofs, I have never done proofs. I have approached my professor for help, and he doesn't seem to help and makes me feel kind of bad for going to him.

My first problem set pertains to Chapter 1 of Spivak's Calculus. I understand all the basic properties, and I understand the proofs we did in class, but I do not understand how to do them on my own. I think I am mostly afraid that I won't do enough steps to prove things. My classmates all seem to understand these things, but I REALLY just need some help getting started.

Sorry I don't have much of an attempt other than staring at my list of the 12 properties and the problem for the past forty five minutes. Where do I begin? I am very discouraged, but I don't want to drop my course.

Thank you so much! I would REALLY appreciate some help.

 

[Dick]

This is hopeless unless you say what the "twelve basic properties" are or give an online reference. Can you prove x^2=y^2 is equivalent to x^2-y^2=0? Can you prove (x-y)(x+y)=x^2-y^2? Is there a property that says a*b=0 implies a=0 or b=0? Or can you prove that from the "twelve properties"?

 

[katia11]

I am very sorry! I didn't know these weren't "universal" twelve properties.

Property 1 a+ (b + c) = (a + b) = c Assosiative Law of Addition
Property 2 a + 0 = 0 + a = a Existence of Additive Identity
Property 3 a + (-a) = (-a) + a = 0 Existence of Additive Inverse
Property 4 a + b = b + a Communitive Law for Addition
Property 5 a X ( b X c) = (a X b) X c Assosiative Law for Multiplication
Property 6 a X 1 = 1 X a = a Existence of a Multiplicative Identity
Property 7 a is not equal to zero, a X (a)^-1 = (a)^-1 X a = 1 Existence of a Multiplicative Inverse
Property 8 a X b = b X a Communtative Law for Multiplication
Property 9 a (b + c) = ( a X b) + (a X c)
Property 10 For every number a, one and only one of the following holds
i) a = 0 ii) a is in the collection P (positive numbers) iii) -a is in the collection P Trichotomy Law
Property 11 If a and b are in P, then a + b is in P Closure Under Addition
Property 12 If a and b are in P, then a X b is in P Closure Under Multiplication

I am so sorry! I actually registered awhile ago but never really needed to post until now. I am just a tad lost.

As to your questions, I cannot prove those first two things. In class we proved that a * 0 = 0, so a * b = 0 should be similar.

I know this is probably a really easy proof but I have never done these outside of class before. Thank you so much!

 

[Dick]

Oh, you don't have to be THAT sorry! No, the 12 properties are not universal. But there are some that aren't really in the list that you can use. They are the properties of 'equality'. If a=b then a+c=b+c. If you use that with a=x^2, b=y^2 and c=-y^2 you've got that x^2=y^2 is the same as x^2-y^2=0. To prove (x+y)(x-y)=x^2-y^2, just use the properties 9 and 3. Um, now for the last step, I'm still looking. Really, you just prove these things by using the algebra you already know and then looking back for the formal justification. And you don't have to prove everything directly from the axioms. Often you can use theorems derived from the axioms that you've already proven.

 

[gabbagabbahey]

I think you can use property 7 (along with the property a*0=0 that you showed in class) to prove the last step by examining two cases;

(1)$a=x+y$ and $a \neq zero$

(2)$a=x-y$ and $a \neq zero$

In both cases just multiply both sides of the equation by $a^{-1}$

Then use property 3 again and your done.

 

[Dick]

Proving a*b=0 implies a=0 or b=0 is a combination of properties 10, 12 and 3. I think you've probably already proved it. Can you check? The details of that are pretty unimportant to what you actually want to prove.

 

[Dick]

I think you can use property 7 (along with the property a*0=0 that you showed in class) to prove the last step by examining two cases;

(1)$a=x+y$ and $a \neq zero$

(2)$a=x-y$ and $a \neq zero$

In both cases just multiply both sides of the equation by $a^{-1}$

Then use property 3 again and your done.

Sure. That works. Thanks. This was making my head spin.