Thread: Prove a Proposition

1 (a) If , and , then .
(b) If , and , then .
(c) If , and , then .
(d) If , then .


Axioms
(1) If and , then their product is in .
(2) , for all .
(3) for all .
(4) contains an element , such that for every .
(5) If and , then there exists an element such that .

So (c) and (d) follows from Axiom (5), and (b) follows from Axiom (2). How about (a)?

Originally Posted by heathrowjohnny

1 (a) If , and , then .
Axioms
(1) If and , then their product is in .
(2) , for all .
(3) for all .
(4) contains an element , such that for every .
(5) If and , then there exists an element such that

(a) follows from axioms (2), (5), (3), and (4), in that order. since xy=xz, you can say that yx=zx using (2) on both sides. you then can show that yx(1/x)=zx(1/x) using (5). this becomes y(1)=z(1) using (3). your equation then becomes y(1)=z(1). using axiom (4) will reduce your equation to the desired y=z.