NAME
CCR 5
MAT 300 · FALL 2014
Critique and correct the following proof, as if you were grading someone’s homework in
great detail. (Write directly on this sheet.) Also provide a revised version of the proof;
your revision should use the basic idea of the original proof (if possible), but should be an
improvement in terms of reasoning and exposition. (Write your proof on the back of this
sheet.)
Theorem. There exists a unique y ∈ R such that for every x ∈ R we have
xy = (xy)2 .
Let x ∈ R be arbitrary. Since xy = (xy)2, we
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Proof.
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have xy = (xy)(xy), so xy = 1, so y = 1/x. Thus there
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exists y such that xy = (xy)2 for every x.
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For uniqueness, suppose also z ∈ R satisfies xz = (xz)2
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for every x ∈ R. Then as above, xz = 1, so z = 1/x.
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Thus z = y, and this shows uniqueness.
Date: October 14, 2014. Due Date: Wednesday, October 22, 2014.
S. Kaliszewski, School of Mathematical and Statistical Sciences, Arizona State University.