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Math Help - Prove a Proposition

  1. #1
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    Prove a Proposition

    1 (a) If  x \neq 0, and xy = xz, then y = z.
    (b) If x \neq 0, and xy = x, then y = 1.
    (c) If x \neq 0, and xy = 1, then y = 1/x.
    (d) If x \neq 0, then 1/(1/x) = x.


    Axioms
    (1) If x \in F and y \in F, then their product xy is in F.
    (2) xy = yx, for all x,y \in F.
    (3) (xy)z = x(yz) for all x,y,z \in F.
    (4) F contains an element 1 \neq 0, such that 1x = x for every x \in F.
    (5) If x \in F and x \neq 0, then there exists an element 1/x \in F such that x \cdot (1/x) = 1.

    So (c) and (d) follows from Axiom (5), and (b) follows from Axiom (2). How about (a)?
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  2. #2
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    Quote Originally Posted by heathrowjohnny View Post
    1 (a) If  x \neq 0, and xy = xz, then y = z.
    Axioms
    (1) If x \in F and y \in F, then their product xy is in F.
    (2) xy = yx, for all x,y \in F.
    (3) (xy)z = x(yz) for all x,y,z \in F.
    (4) F contains an element 1 \neq 0, such that 1x = x for every x \in F.
    (5) If x \in F and x \neq 0, then there exists an element 1/x \in F such that x \cdot (1/x) = 1
    (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.
    Last edited by xifentoozlerix; February 16th 2008 at 02:37 PM. Reason: forgot a step
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