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Let $$a = b$$ $$c = d$$

Assume a,b,c,d are non-zeros

If we know that $$\frac{a}{c} = \frac{b}{d}$$

Are the original equations valid? I know we can arrive at a true statement from false statements so even if original equations are not valid, the derived equation can be valid, right? I'm still confused tho because there is not really any logical mistakes in steps that could lead to a true statement from two false statements in this case.

Thanks in advance.

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1 Answer 1

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$\frac{1}{2} = \frac{2}{4}$ but $1 \neq 2$ and $2 \neq 4$.

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  • $\begingroup$ So there is not really any way to know if original equations are valid from derived equation? (always) $\endgroup$
    – AmirWG
    Commented Feb 17, 2022 at 1:32
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    $\begingroup$ @AmirWG In general, yes. Basically if we know $p \implies q$ and $q$ is true, we still know nothing about $p$. $\endgroup$
    – Clement Yung
    Commented Feb 17, 2022 at 1:35
  • $\begingroup$ got it, thanks alot. $\endgroup$
    – AmirWG
    Commented Feb 17, 2022 at 1:38

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