help We will assume the following facts1 about these operations: Things to be pr
ID: 2987280 • Letter: H
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We will assume the following facts1 about these operations: Things to be proven: 0 + 0 = 0 Idempotency of the identity. 2. a x y, if a + x = a + y, then x = y. Cancellation. a, if a + a = a, then a = 0. Uniqueness of idempotent. a !b such that a + b = 0. Uniqueness of inverse. The symbol ! reads, there is exactly one. One proves uniqueness usually by taking two things that satisfy something, and proving that they are actually equal. Definition. The unique b in 4 is then called -a, the negative of a. Define subtraction by a - b = a + (-b).Explanation / Answer
1.
0 is identity so a+0 = a for all a, in particular for a = 0, so we have 0+0 = 0.
2.
there exists b such that a+b = b+a = 0 (A4 and A1)
a+x = a+y
=> b+(a+x) = b+(a+y)
=> (b+a)+x = (b+a)+y (A2)
=> 0+x = 0+y (since b+a = 0)
=> x = y (A3 and A1 give 0+a = a for all a)
3.
There exists b such that b+a = 0 (A4 and A1)
a+a = a
=> b+(a+a) = b+a
=> (b+a)+a = b+a (A2)
=> 0+a = 0 (since b+a = 0)
=> a = 0 (A3 and A1 give 0+a = a for all a)
4.
suppose a+b = 0 = a+c
Then we have a+b = a+c,
thus by cancellation which we proved in part 2, we have b=c.
Thus there exists unique b such that a+b = 0 (existence from A4 and uniqueness from this part)
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