Prove A & B Using Axioms & Definitions: Axiomatic Approach 3

[solakis1]

Given the following axioms:

For all a,b,c we have:

1) a+b = b+a
2) a+(b+c)=(a+b)+c
3) ab = ba
4) a(bc) = (ab)c
5) a(b+c) =ab+ac
NOTE,here the multiplication sign (.) between the variables have been ommited
6) There ia a number called 0 such that for all a,
a+0 =a
7)For each a, there is a number -a such that for any a,
a+(-a) = 0
8)There is a number called 1(diofferent from 0) such that for any a,
a1 = a
9)For each a which is different than 0there exists a number called 1/a such that;
a.(1/a)= 1.

10) exactly one of a>b,b>a or a=b holds
11) if a>b ,b>c then a>c
12) if c>0 ,a>b then ac>bc
13) if a>b then a+c>b+c for any c

The definitions:

14) a/b = a(1/b)

15) $a\geq 0\Longrightarrow |a|=a$ and $ a<0\Longrightarrow |a|=-a$.

Then by using only the axioms and the definitions above prove:A) If a>0 and b>0 then a>b iff aa>bbB) If $x\neq 0$ then (|x||x|)/x=x for all x

 

[I like Serena]

Hi solakis,

What have you tried?

 

[solakis1]

Hi solakis,

What have you tried?

Hi

A is easy but for B the 2nd part i get stuck

The part were we consider x<0

And another thing

Why should we not consider the definition of absolute value as an axiom and not as a definition

 

[I like Serena]

Hi

A is easy but for B the 2nd part i get stuck

The part were we consider x<0

I guess you mean how to prove that (-x)(-x)=xx?

Perhaps you can prove that (-x)=(-1)x?
Or that (-1)(-1)=1?

And another thing

Why should we not consider the definition of absolute value as an axiom and not as a definition

Because it's a notation.
An axiom would be if the notation is used in a statement.
Similarly (1/x) and (-x) are notations or definitions to identify the multiplicative inverse respectively the additive inverse.

The axiom is that for every x there exists an additive inverse, denoted as (-x), such that x+(-x)=0.
This is a statement.

 

[solakis1]

I guess you mean how to prove that (-x)(-x)=xx?

Perhaps you can prove that (-x)=(-1)x?
Or that (-1)(-1)=1?

Because it's a notation.
An axiom would be if the notation is used in a statement.
Similarly (1/x) and (-x) are notations or definitions to identify the multiplicative inverse respectively the additive inverse.

The axiom is that for every x there exists an additive inverse, denoted as (-x), such that x+(-x)=0.
This is a statement.

-,/ are not notations ,but two of the primitive symbols about which we write the two axioms:

a+(-a)=0
$a\neq 0\Longrightarrow a\frac{1}{a} = 1$.

Or in the language of the predicate logic one place operation terms.

Who said that | | cannot be taken as a one place operation symbol and consider as a primitive in our axiomatic system.

Is it not the :

$x\geq 0\Longrightarrow |x|=x$ a statement?

And another thing ,if the above plays the same role in a proof as an axiom why called definition

 

[I like Serena]

-,/ are not notations ,but two of the primitive symbols about which we write the two axioms:

a+(-a)=0
$a\neq 0\Longrightarrow a\frac{1}{a} = 1$.

notation equals primitive symbol.

And another thing ,if the above plays the same role in a proof as an axiom why called definition

We call it a definition if it is about a word or a symbol (or set of symbols) that has no meaning yet until we give it a definition.
A definition typically has the form "<word> is defined as <definition>", or "$|\cdot|$ is defined as <definition>", although the exact form can vary.

We call it an axiom if it's a statement making use of the words and symbols we defined, that is accepted as implicitly true.But hey, you don't have to take my word for it if you think you know better.