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Let $p, q, r$ be mathematical statments.

Suppose we know:

  • "$p$ and $q$" $\Rightarrow$ "$r$" is true;
  • "$r$" $\Rightarrow$ "$q$" is true.

Is "$p$" $\Rightarrow$ "$r$" true?

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up vote 10 down vote accepted

Not necessarily, because perhaps $p$ is true, and $q$ and $r$ are false. In this case, both of your implications come out true (note that $p$ and $q$ implies $r$ is true vacuously, since the hypothesis is false, and similarly for $r$ implies $q$), but the final implication $p\implies r$ comes out false.

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Let $q < 1$ and $q < -1$

$$ \begin{align} q < 1 \wedge q < -1 &\Rightarrow q < -1 &\text{(True)}\\ q < -1 &\Rightarrow q < -1 &\text{(True)}\\ q < 1 &\Rightarrow q < -1 & \text{(False)} \end{align} $$

I'm certainly not a professional mathematician, but this is how I would interpret it.

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May be

$p$ is $T$
$q$ is $F$
$r$ is $F$

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