A question regarding propositional logic Good day, I'm currently studying for an exam and need to learn about propositional logic. Well, since I'm not good at English I'll just write what I've done so far:
$(A \land (B \rightarrow \neg A)) \rightarrow \neg B$
I started like this:
$(A \land (\neg B \lor \neg A)) \rightarrow \neg B$
$\neg (A \land (\neg B \lor \neg A)) \lor \neg B$
$\neg A \lor \neg (\neg B \lor \neg A)) \lor \neg B$
$\neg A \lor (B \land A) \lor \neg B$
$\neg A \lor \neg B \lor (B \land A) $
And I think it is the same as:
$( \neg A \lor A) \land (\neg B \lor B) $
Am I on the right road? Because I'm really not sure anymore. However, I think this is a tautology.
 A: Well, in this style of argument, why not proceed from your fourth line
$$\neg A \lor (B \land A) \lor \neg B$$
by rearranging
$$(B \land A) \lor \neg B \lor \neg A$$
whence
$$(B \land A) \lor \neg(B \land A)$$
which is of the form
$$\varphi \lor \neg\varphi$$
and hence a tautology. So the original wff is a tautology as you wanted to show.
A: You’re approaching it in a reasonable way. Essentially the same approach can be organized a bit more clearly. I’ll start by simplifying part of the expression:
$$\begin{align*}
A\land(B\to\neg A)&\equiv A\land(\neg B\lor\neg A)\\
&\equiv(A\land\neg B)\lor(A\land\neg A)\\
&\equiv(A\land\neg B)\lor\bot\\
&\equiv A\land\neg B\;,
\end{align*}$$
where $\bot$ is a symbol for a contradiction, something that’s always false; you may be accustomed to writing F or the like instead. Then
$$\begin{align*}
\big(A\land(B\to\neg A)\big)\to\neg B&\equiv(A\land\neg B)\to\neg B\\
&\equiv\neg(A\land\neg B)\lor\neg B\\
&\equiv(\neg A\lor\neg\neg B)\lor\neg B\\
&\equiv(\neg A\lor B)\lor\neg B\\
&\equiv\neg A\lor(B\lor\neg B)\\
&\equiv\neg A\lor\top\\
&\equiv\top\;,
\end{align*}$$
where $\top$ is a symbol for a tautology, something that is always true.
Note that you can always determine whether a propositional expression is a tautology by using a truth table; this can be a useful check even when you’re required to produce an algebraic argument.
