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Why isn't this valid?

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?

Reply 1

Anon the 7th

Simplicity

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

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)

Reply 2

username219321

Because to get from the " 0a = 0b" to "a = b" stage you are dividing by zero.

Reply 3

SimonM

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