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On the internet there are many proofs but very summary

Definition
$|a| = $
$a$ if $a \ge 0$
$-a$ if $a < 0$

Proposition 1
$|a| = 0 \Leftarrow \Rightarrow a = 0$

$\Rightarrow$)

Suppose that $|a| = 0$
This implies that $a\ge 0$ xor $a<0$

Case $a<0$
$|a| = -a$ by hypothesis $-a= 0$
As $a<0$
$a + (-a)<0 + 0$
$0<0$ (!contradiction)

Case $a \ge 0$
$|a| = a$ by hypothesis $a = 0$


$\Leftarrow$) Suppose that $a= 0$
$|a| = a$ by hypothesis $|a| = 0$

proposition 2
$|a| \ge 0$

case $a \ge 0$
$|a| = a$ by hypothesis $|a| \ge 0$

case $a < 0$
$|a| = -a$
As $a<0$
$a+(-a)<0+(-a)$
$0<-a$
$0<|a| \Rightarrow 0 \le |a|$

are valid my proofs?

Jose Vega
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1 Answers1

1

I think you do a little bit complicate. $$|a|=0\iff (a=0\quad \text{or}\quad -a=0)\iff a=0.$$

Moreover, you don't need that fact that $a\geq 0$.

Surb
  • 55,662
  • It is strange that you would have BOTH "a= 0" and "a>= 0". a= 0 is certainly included in a>= 0 but just a= 0 is more precise. – user247327 Feb 05 '16 at 17:36
  • @user247327: I don't have both ! I actually already made the remark to the OP that he doesn't need $a\geq 0$. Could you give more precision on your comment ? – Surb Feb 05 '16 at 17:39
  • I am sorry maybe I was not clear, $|a| = 0$ iff $a = 0$ is a proposition and $|a| \ge 0$ is other proposition – Jose Vega Feb 05 '16 at 17:50
  • which side? (I don't understand you) – Jose Vega Feb 05 '16 at 22:04
  • Dear @JoseVega: if $a\geq 0$, then $|a|=a\geq 0$. If $a\leq 0$, then $|a|=-a\geq 0$. finish :-) – Surb Feb 05 '16 at 22:08
  • @Surb if $a \le 0$, then $|a| = -a \ge 0$ I disagree because the target is $a=0$ – Jose Vega Feb 05 '16 at 22:18
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    So may be you can write an unconfused assertion ! What I read is $$\ \ $$ 1) prove that $a=0\iff |a|=0$. $$\ \ \ $$2) Prove that $|a|\geq 0$. $$\ \ $$ Anyways, it's not important now, I quite ! Ask somebody else. Good evening. – Surb Feb 05 '16 at 22:20