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I know the question has already been asked here but both answers it got used something they call "the transitive property of inequality" and "the subtractive property of inequality". I got this task from Chrystal's Algebra: An Elementary Textbook for the Higher Classes of Secondary Schools and for Colleges fifth edition, Part I, right at the end of page $10$ and beginning of page $11$. Right after he defines $0$ as $+a-a=-a+a$, he says we derive the result $b+0=b=b-0$ and $+0=-0$. And he leaves this last two results to be proved "by applying the laws of commutation and association along with the definition of $0$". I have worked on it for a time now and I have not been able to reach an acceptable proof by using what he asks to use for it.

So far I have gotten this: $\begin{align}+0&=+0-0=-0+0\quad\text{By the definition of $0$}.\\ & =-0+(+0-0)\qquad\,\text{By the definition of $0$ again and the associative law.}\\ &=-0+0-0=-0 \quad\text{By the relationship of subtraction and addition.}\end{align}$

(This last step is the one that does not convince me because I used something else that was not specified in the problem. Anyways he had used that definition before to work on other problems.

I am pretty confident that if you read the first ten pages of that book you will know why one should not use other results as in the other question I cited (that is, the transitive property of inequality). I sincerely apreciate your help

Robert Howard
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Felmola
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    How is $-a$ defined before $0$? How does the author show that his $0$ is well-defined? – Hagen von Eitzen Dec 08 '17 at 17:50
  • Here's the question in its context within the textbook: https://books.google.com/books?id=ZoC4AAAAIAAJ&pg=PA10 – Chris Culter Dec 08 '17 at 18:13
  • That's a pretty old -fashioned book. If you're reading it for historical reasons, more power to you. If you want to learn contemporary abstract algebra and proof techniques there are more modern books. – Ethan Bolker Dec 08 '17 at 22:29
  • @HagenvonEitzen I truly do not know. In page 5 he defines subtration as the inverse of addition: _-b annuls the effect of +b and vice versa...Thus, - is defined relatively to + by the equation:

    +a-b+b=+a. +a+b-b=+a_.

    The first mention of 0 comes in the last part of page 10 as I mentioned in the question above

     – Felmola Dec 09 '17 at 03:10

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