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Ok this should be a rather simple question but I don't know if the key is wrong here or I if I'm wrong.

The question is: Determine all true implications/logical equivalence of the following statemtens:

A: $a=b$

B: $a^2=b^2$

C: $ab=b^2$

I said $A\implies B$ and $A \iff C$ but the answer key says $A \implies B$ and $A \implies C$

I don't understand. If I'm wrong here I have no clue to why since you can change the C statement into A by dividing by b on both sides. Please help me answer this, what is suppose to be, simple question.

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Suppose $a=1$ and $b=0$. Then $C$ is true, but $A$ is false, so we don't have $C \implies A$.

However, on a set of numbers that doesn't contain $0$, we have $A \iff C$.

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  • $\begingroup$ The proposer's error was "you can...divide by b". Sometimes you can't because b may be zero, especially on Tuesdays. $\endgroup$ – DanielWainfleet Oct 4 '15 at 23:48
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Cancellation law: $ab=b^2$ only if $b\neq0$ so, $C\Rightarrow A$ only if $b\neq0$

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