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:
- If x+y = x+z, then y=z
- If x+y =x, then y = 0
- If x+y = 0, then y= -x
- -(-x) = x
Eg, my teacher has proved the first statement by doing this:
- 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.