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our new maths teacher (BSc Hons) just started a topic. He wrote all these theorems but I'm unable to understand them and prove the statements. They are as follows:

Theorem: The addition axioms for field imply the following statements:

  1. If x+y = x+z, then y=z
  2. If x+y =x, then y = 0
  3. If x+y = 0, then y= -x
  4. -(-x) = x

Eg, my teacher has proved the first statement by doing this:

  1. If x+y = x+z, then y=z

Y= 0+Y

=(-x+x) + y (Inverse)

=-x + (x+y) (Associative)

=-x + (x+z) (Given)

=(-x+x) + z (Inverse)

=(0) + z (Identity)

=z

I've tried so hard to understand, can anyone give me an idea how to solve them? If you want, I can write all of the proofs if anyone is willing to make me understand them.

Lord Asbaat
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    What specifically are you having trouble understanding about them? Are you having trouble understanding the first theorem's proof which your professor already gave you as well? Have you considered using the first theorem to prove the second theorem noting that $x=x+0$ by the definition of $0$ and letting $0$ play the role of $z$ for the use of the first theorem? – JMoravitz Mar 21 '17 at 15:55
  • In addition to the hint from @JMoravitz, notice that item 1 also immediately implies item 3 by letting $-x$ play the role of $z$. – Andreas Blass Mar 21 '17 at 16:05
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    I've never studied these axioms in my whole life. The trouble I'm having is that how do I shape the answer. For example,I would never have thought of the step y = 0+y to prove it. Can you give me a link or a good book which contains the explanation of these theorems? – Lord Asbaat Mar 21 '17 at 16:17
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    I don't know that there is a good answer to that... just learn what the axioms are by heart and keep them in your toolbox of available things to use. This isn't like the math you learn in elementary school where its just computation, this is more like a puzzle where you fit together pieces the right way to make the bigger picture. Your saying "I would never have thought of the step y=0+y" is to me like saying "I never would have thought to twist the puzzle piece to make it fit". Certainly, sometimes which move to use is not so obvious, but the moves available to you should all be known. – JMoravitz Mar 21 '17 at 16:41
  • Thanks, I think you perfectly read my mind. I'll need to learn them by heart and try to solve this puzzle. Thanks, I'll try to do it once again! – Lord Asbaat Mar 22 '17 at 12:36

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