Prove $-(-a) = a$

Question

Let $F$ be a field and $a \in F$. Prove $-(-a) = a$.

So we want to show that $(-a) + (-(-a)) = 0$, since inverses are unique (I successfully proved that inverses are unique in an earlier problem which was marked correct). I added $a$ to both sides, and got: $a + ((-a) + (-(-a))) = a + 0$, so using associativity and commutativity, we have $((-a) + a) + (-(-a)) = 0 + a$ which is $0 + (-(-a)) = a$, and so $(-(-a)) = a$. But this was marked incorrect. What is the proper proof for this property?

Answer 1

Your answer is essentially correct. I'd write the same answer as:

$$\begin{align} a &= a+0 \\&= a+((-a)+(-(-a))) \\&= (a+(-a))+(-(-a)) \\&= 0+(-(-a)) \\&= -(-a) \end{align}$$

So we are applying the associative law and the fact that $x+(-x)=0$ for $x=a$ and $x=-a$. and $0+x=x+0=x$.

Note, you converted:

$$a+((-a)+(-(-a))) = ((-a)+a) + (-(-a))$$

which is a confusing combination of associativity and commutatitivity. Better to apply one at a time.

Answer 2

Your proof is correct, but I think you are using "many" steps.

Hint. Define $b:=-a$ and use $b+(-b)=0$.

Examining the equation $a + (-a) = 0$ from a different point of view. For we have also $-(-a) + (-a) = 0$; by the cancellation law we obtain $-(-a) = a$.