I need to write the following statements into logical form and determine whether the conclusion is logically implied by the assumptions.
A sufficient condition of $f$ to be integrable is that $g$ be bounded. A necessary condition for $h$ to be continuous is that $f$ is integrable. Hence, if $g$ is bounded or $h$ is continuous, then $f$ is integrable.
This is my work:
$p$: $f$ is integrable
$q$: $g$ is bounded
$r$: $h$ is continuous
$((q\rightarrow p)\wedge (p\rightarrow r))\rightarrow ((q\vee r)\rightarrow p)$
Is this the correct form? I don't think it is, because it should be a tautology and yet it is not...right?
No, the second claim should be $r \to p$: $f$ being integrable $p$ is a necessary condition, rather than a sufficient condition. And when something $p$ is a necessary condition for something else $q$, you translate it as $q \to p$
Also, I much prefer to present these statements as three separate statements as part of an argument, rather than as one big conditional statement, so I would symbolize this as :
$q \to p$
$r \to p$
$\therefore (q \lor r) \to p$