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I need to prove the following property, but I don't know how: $$-(-x)=x.$$

Please help me. Thanks for your attention.

user100991
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$-(-x)=-(-x)+0=-(-x)+x+(-x)=[-(-x)+(-x)]+x=0+x=x$

Madrit Zhaku
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    if you swap $x + (-x)$ by $(-x) + x$ in the second step, you don't even need commutativity. – AlexR Oct 16 '13 at 19:17
  • dear college, $x+(-x)=(-x)+x=0$, I am sure you understand what I've done – Madrit Zhaku Oct 16 '13 at 19:19
  • I know, just a bit cleaner this way. All fine anyways, mate. Also, some Groups don't need this (i.e. talk about left-inverse / right-inverse) – AlexR Oct 16 '13 at 19:20
  • This property requires a simple confirmation, and I believe that he has given this example requires this thing, be shown in a simple way – Madrit Zhaku Oct 16 '13 at 19:23
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Assuming you meant in a group (or ring or field) and you're working axiomatically : use the uniqueness of additive inverse.

By definition, $\;-x\;$ is the unique element that fulfills $\;x+(-x)=0\;$ and the other way around, meaning: $\;x\;$ is the unique additive inverse of $\;-x\;$ , but then

$$-(-x)+(-x)\stackrel{\text{distributivity}}=(-1+1)(-x)=0\cdot x\stackrel{\text{hopefully you already proved this!}}=0$$

The above means that $\;-x\;$ is additive inverse also of $\;-(-x)\;$ , and from here that uniqueness forces $\;x(-x)=x\;$

DonAntonio
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  • Why force a Ring by using multiplication?!? – AlexR Oct 16 '13 at 19:26
  • Seeing the level of he question most probably it is about the reals. – DonAntonio Oct 17 '13 at 04:21
  • Still an unnecessary assumption. Seing he has to prove this, it's more likely about Groups in general (maybe abelian). But nvm. – AlexR Oct 17 '13 at 09:12
  • Perhpas in your experience @AlexR , not in mine. Here first year students must learn the basic axioms of fields to work their way into basic linear algebra. Much later will come groups and rings . – DonAntonio Oct 17 '13 at 10:09
  • Well, over here students are confronted with the axiomatics of groups in linear algebra and this is expanded to rings. Proving the OP's assignment naturally emerges in the part about groups :) – AlexR Oct 17 '13 at 10:15
  • Also here students are taught the basics of group and righ theory in Linear Algebra I... at the end of the course . At the beginning they're told about fields only to make, I'm sure, their way into the abstract part of it smoother. And what the OP asked belongs exactly to the part of fields. After this is trivial to exapand it to other algebraic structures. – DonAntonio Oct 17 '13 at 10:17
  • I see what you mean. I'd remove my DV but unfortunately it's locked by now. – AlexR Oct 17 '13 at 10:19
  • Never mind, though if you had followed my history in this site you'd see what my opinion on DV are. – DonAntonio Oct 17 '13 at 10:20