Proof of X - (-A) = X + A. (Is it possible to prove?)

Question

I have a problem with arithmetics. Is there is a proof that X - (-A) = X + A? Not just some analogy or "mind-trick". If so, could you tell me it or at least give some link or refer me somewhere? (may be a book, video, some website or whatever that might be helpful). Thank you very much. (by the way, let's be friends, may be?..)

Answer 1

I will assume that in $X\color{red}{-}(\color{blue}{-} A)$, the red $\color{red}{-}$ is being used to denote subtraction, which is defined as $X\color{red}{-}A = X+(\color{blue}{-}A)$ while the blue $\color{blue}{-}$ is being used to denote additive inversion, that is to say $(\color{blue}{-}A)$ is the additive inverse of $A$.

Now... given a number $A$, the additive inverse of the number $(\color{blue}{-}A)$ is a number with the special property that $A+(\color{blue}{-}A)=0=(\color{blue}{-}A)+A$.

Further, we can see that each number has exactly one additive inverse since otherwise if both $\color{blue}{-A}$ and $\color{green}{-A}$ were both potentially different additive inverses of $A$ then we would have $(\color{blue}{-A})=(\color{blue}{-A})+0=(\color{blue}{-A})+A+(\color{green}{-A})=0+(\color{green}{-A})=(\color{green}{-A})$, so we get that additive inverses must be unique.

Finally, we realize then that since additive inverses must be unique, then the additive inverse of the additive inverse of $A$ must be $A$ itself. That is to say, $(\color{blue}{-}(\color{blue}{-}A))=A$

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Putting all this together, we have that $X\color{red}{-}(\color{blue}{-}A)=X+(\color{blue}{-}(\color{blue}{-}A))=X+A$

Answer 2

Assuming two standard definitions:

(i) $-x$ is the unique number such that $x+(-x)=0$,

(ii) $x-y=x+(-y)$:

By definition $X-(-A)=X+(--A)$, so it's enough to show that $--A=A$. By (i) this is the same as $-A+A=0$; but again by (i), This follows from (i): $$-A+A=A+(-A)=0.$$

Answer 3

Let $a \in \mathbb{R}$.

The additive inverse of $a$ is $(-a)$.ie.

$a+(-a)=0$.

One writes $x-a$ for $x+(-a)$, $x$ real.

Want to show:

$x+(-(-a))= x+a.$

Note the additive inverse of $(-a)$ is $(-(-a))$.

$x+(-(-a))+ 0=$

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

$x+[(-(-a))+(-a)] +a =$

$(x +0)+a= x+a$.

Used:

$y+0=y$, $y$ real;

Associative law of addition.