Question

Using the following axioms:

a.) (x+y)+x = x +(y+x) for all x, y in R (associative law of addition)

b.) x + y = y + x for all x, y elements of R (commutative law of addition)

c.) There exists an additive identity 0 element of R (x+0 = x for all x elements of R)

d.) Each x element of R has an additive inverse (an inverse with respect to addition)

Prove the following theorems:

1.) The additive identity in R is unique.

2.) Every real number x has a unique additive inverse

3.) x+y = x + z if and only if y = z, assuming that x, y, and z are elements R

4.) Suppose x,y are elements of R. Then the following two statements hold

(i) If x+y = x, then y = 0

(ii) If x + y = 0, then y = -x

Answer 1