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Distributive Law in a Field

  1. Sep 6, 2011 #1
    Suppose we're in a field F in which x,y,z are members. Consider the axiom of distribution which states that

    xy + xz = x(y + z) for all x,y,z ∈ F.

    Now consider this deduction:
    xy + xz - xy ⟹ xy + x(z-y)

    True no doubt, but to check my understanding aren't we missing a step? Shouldn't we more accurately say:
    xy + xz - xy ⟹ xy + xz + x(-y) ⟹ xy + x(z-y)

    I'm trying to self-study Rudin and just want to check my understanding.
     
  2. jcsd
  3. Sep 6, 2011 #2
    Yes, you are correct. And we could even be more accurate and say

    xy+xz-xy=xy+xz+x(-y)=xy+x(z+(-y))=xy+x(z-y)
     
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