How can I show that three statements are not logically equivalent to another? I am given three premises and a conclusion. The premises are:
\begin{gather}
p \lor q \\
p \to \mathord{\sim}q \\
p \to r
\end{gather}
and the conclusion is
$$ r $$
I used a truth table and showed that this is logically incorrect. However, how would I go about this without a truth table?
 A: 
"How would I go about this without a truth table?"

To show that some propositional-calculus premisses $A, B, C$ do not entail some conclusion $D$, you need to find a counter-valuation, i.e. valuation to the atoms in the premisses and conclusion which makes the premisses true and conclusion false.
Now, doing a truth-table is a systematic search through possible valuations of the atoms. So if an argument is invalid, if you start off doing a full truth-table, you must eventually hit on a countervaluation which makes the premisses true and conclusion false. That's what's so great about truth-tables: they must deliver the goods!  But also, that's what's so laborious about them, churning through line after line!!
Can you speed things up?
Well, in the present case you can try random guess work, and with some luck hit on the valuation $p \Rightarrow F$, $q \Rightarrow T$, $r \Rightarrow F$ which shows the inference to be valid.
But better, much better, you can "work backwards". Don't make blind guesses, but ask: what would a countervaluation which makes the  premisses $A, B, C$ true sand $D$ false look like? In the present case, entail some conclusion $D$ $p \lor q$ , $p \to q$ , $p \to r$ all true, and the conclusion $r$ false look like?
Well, you know already that you must have $r \Rightarrow F$. But then, to make $p \to r$  true that means $p \Rightarrow F$. But then to make $p \lor q$ and  $p \to q$ true we need $q \Rightarrow T$. Bingo!
Now there's a lovely way of systematising that "working backwards" method of homing in on a counter valuation for an invalid inference -- it's called the tableau method. And if you systematically work backwards and fail to find a countervaluationm that shows the argument is valid. Many textbooks explain this method (including, ahem, P*t*r Sm*th's Introduction to Formal Logic which I'm told is quite accessible ....!).
A: The assignments $p=r=\text{false},\; q=\text{true}$ renders all the premises true, but the conclusion false, with no contradiction. Therefore, the argument is invalid. 
