If "$p \mathrel{\&} q \Rightarrow r$" and "$r \Rightarrow q$" are true, is "$p \Rightarrow r$" also true?

Question

Let $p, q, r$ be mathematical statements. Suppose we know:

• "$p \mathrel{\&} q \Rightarrow r$" is true; and
• "$r \Rightarrow q$" is true.

Is "$p \Rightarrow r$" true?

Answer 1

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.

Answer 2

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.

Answer 3

May be

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