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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?..)

Tim Solnze
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    If you want a real proof and 'not just some analogy or "mind trick"', you will have to tell us exactly how you define $+$, $-$ and $-$. (Yes, those are two $-$ signs. They are different.) Part of that will probably include how you define the numbers that $X$ and $A$ are part of (natural numbers, real numbers, or something else). Without formal definitions we can't prove anything (this goes for math in general). – Arthur May 25 '19 at 14:23
  • What do you mean with $-A$, $X+A$ and $X-A$? We need to know your perception of that. – drhab May 25 '19 at 14:23
  • Look at https://math.stackexchange.com/questions/137695/why-minus-times-minus-needs-to-be-plus – The Pheromone Kid May 25 '19 at 14:24
  • whoah guys, yeah, that's harder than I thought. X and A are a part of real numbers. -A is the negative of A, that is , the inverse of A. So minus sign(the first one) means that I substract one number from another. I don't know how to define that properly.. perhaps I can say that if z - 1 = b, then b is the precursor of z(to the left on the numberline). If z - 2 = k, then k is the precursor of the precursor of z(or the precursor of b, if consider previous sentence). Now, if x + 1 = L, then L is the successor of x(right on the numberline). but now I have a problem, since it is real numbers – Tim Solnze May 25 '19 at 14:36
  • You should draw yourself a number line, going from, let's say $-10$ to 10. Then try on it a few simple examples, like $3 + 7$, $3 - 7$, $3 - (-7)$, etc. – Robert Soupe May 25 '19 at 15:48
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3 Answers3

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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): $$-A+A=A+(-A)=0.$$

David C. Ullrich
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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, $(\color{blue}{-}(\color{blue}{-}A))=A$

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

JMoravitz
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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.

Peter Szilas
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  • Thank you. One little question. Can you recommend some source on fundamentals of arithmetic? For example, in many textbooks there's no issue on why, say -x * -z = c. (I know why). – Tim Solnze May 26 '19 at 12:24
  • Tim. The above kind of proof is done in the first chapters of Real Analysis books. Real numbers are introduced satisfying axioms, and the above is done a bit in this way. If you want arithmetic , just handling numbers, fractions, and what not , I would not know.Perhaps Google a bit.Hope this helps? – Peter Szilas May 26 '19 at 14:17
  • Hi, I'm sorry. How did you learned logic? Symbolic logic. I became very sad about myself because of logic. It's hard. And there no one who knows the answer to my question. I do not understand truth table for condtitionals and I stopped there. It does not make sense to me. It seems I have to learn logic to proceed with set theory and then with mathematics. How did you learned logic? Can you help me somehow, like refer me somewhere? Thank you. – Tim Solnze Jun 03 '19 at 14:07
  • Tim.Firstly do not give up! I myself am not an expert in formal logic, had a class once, and did not like it a lot.The principles that you need in, say Introductory analysis, you can pick up without a full course in logic. I would start with Introductory Analysis, check out, for example, Serge Lang's book, lots of worked examples you find in Schaum's Outline. Your thoughts ? – Peter Szilas Jun 03 '19 at 14:25