Prove a=a^+-a^- & |a|=a^++a^- for Easy Analysis Proof Homework

[iamalexalright]

Homework Statement

The positive part of an a in R is defined by:

$$a^{+} = (|a| + a) / 2$$

and the negative by:

$$a^{-} = (|a| - a) / 2$$.

Prove that $$a = a^{+} - a^{-}$$ and $$|a| = a^{+} + a^{-}$$

Homework Equations

The field axioms(closure, associativity,...)
The order axioms
Definition of the absolute value (and a few theorems)

The Attempt at a Solution

Now, I'm new at proof writing, but this seems too simple (and I don't know if providing all these steps is too much or not?) : /

$$a = a^{+} - a^{-} <br /> = (|a| + a)/2 - (|a| - a)/2 <br /> = 2^{-1}(|a| + a) - 2^{-1}(|a| - a) <br /> = 2^{-1}(|a| + a - |a| + a) <br /> = 2^{-1}(2a)<br /> = a$$


(btw, how can I separate my proof line by line with LaTeX?)

 

[Mark44]

It looks like you have proved that a = a, which is obvious to the most casual observer, and for this reason, not very interesting.

For the first part, start with a⁺ - a^- and use your definitions. What do you get?
For the second part, start with a⁺ + a^-. What do you get this time?

For your last questions, use separate LaTeX lines.

 

[iamalexalright]

alright, so we'd have something like this:

if $$a \geq 0$$ then $$|a| = a$$
$$(|a| + a) / 2 - (|a| - a) / 2 =$$
$$= (a + a) / 2 - (a - a) / 2 =$$
$$= 2(a)/2 - (0)/2 =$$
$$= a$$

if $$a < 0$$ then$$|a| = -a$$
and then the same procedure

 

[Mark44]

alright, so we'd have something like this:

if $$a \geq 0$$ then $$|a| = a$$
$$(|a| + a) / 2 - (|a| - a) / 2 =$$
$$= (a + a) / 2 - (a - a) / 2 =$$
$$= 2(a)/2 - (0)/2 =$$
$$= a$$

if $$a < 0$$ then$$|a| = -a$$
and then the same procedure

This is correct, but is more complicated than it needs to be.

a⁺ - a^-
= (1/2)(|a| + a) - (1/2)(|a| - a)
= (1/2)[|a| - |a| + a -(-a)]
= (1/2)(2a) = a
This is true for any real value of a, not just for a >= 0.

You should be able to do something similar for the other part.

 

[iamalexalright]

Alright, thanks Mark44. I included it just because I'm not used to writing proofs yet (ie I don't exactly know what to include/omit yet).

 

[Mark44]

Include everything that needs to be there, and omit all the rest. Start at one side of the equation you are trying to prove. Use definitions to rewrite the expression on that side and work toward making what you start with look like the other side you're trying to get to.