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    0=0 \Rightarrow 0 \times a = 0 \times b \Rightarrow a=b

    As it seems like a pretty valid form of reasoning. Why, can't you do that. Certainly, it would have came in handy during a show question in FP2.

    Yeah, so whats wrong with that?

    P.S. Can it be made valid? Is there something in set theory or category theory or logic that stops this?
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    (Original post by Simplicity)
    0=0 \Rightarrow 0 \times a = 0 \times b \Rightarrow a=b

    As it seems like a pretty valid form of reasoning. Why, can't you do that. Certainly, it would have came in handy during a show question in FP2.

    Yeah, so whats wrong with that?

    P.S. Can it be made valid?
    because anything multiplied by 0 equals 0 and so it doesnt technically mean a and b are the same value, however they COULD be the same value but theres no way to see for sure (not that i can think of anyway)
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    Because to get from the " 0a = 0b" to "a = b" stage you are dividing by zero.
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    Why dividing by zero is an issue is depends on what 'thing' you are working over. Lets assume it's a (commutative to make the proof shorter) ring for now.

    Suppose we have a ring where we can divide by zero, that is \exists 0^{-1} such that 00^{-1} = 1

    Lemma: 0a = 0

    Proof: a0 = a(0+0) = a0+a0 \Rightarrow a0=0

    Therefore 0=1

    From this we can conclude that every element of our ring is zero, (the zero ring). So we have a problem when we want to work with larger (more interesting) rings
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    There was me thinking it was just because it involved division by 0
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    Well... you can divide by 0, in the zero ring. But that's only because 0 = 1.
 
 
 
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Updated: August 9, 2009
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