I'm trying to do some past paper questions for revision and find myself perplexed on some of the expressions that need normalized/simplified which involves an implies.
For example:
(A ∧ ¬B) → B ∨ C ∨ ¬ (A ∧ ¬C)
Now I know that A → B can be normalized to ¬A or B, but I can't seem to find an example when it comes to multiple things implying something to learn from. I'd appreciate if someone could explain to me how I would simplify such an expression.
On
I assume that $\to$ is the main connective. First, eliminate $\to$ in favor of $\lor$ and $\neg$, then use De Morgan to eliminate $\land$, and then remove repeated terms: $$\begin{align} (A\land \neg B) \to B\lor C\lor \neg(A\land \neg C) &\iff \neg(A\land \neg B) \lor B\lor C\lor \neg(A\land \neg C) \\ &\iff \neg A\lor B \lor B\lor C\lor \neg A\lor C \\ &\iff \neg A\lor B \lor C.\\ \end{align}$$ If you wish, you can now reintroduce $\to$, yielding the equivalent formula: $$ A\to (B \lor C). $$
The rule that you cite:
$$A \rightarrow B = \neg A \vee B$$
also works when $A$ is a compound expression. (All of these rules do). In your example, the simplification would go like this:
$$ \begin{aligned} (A \wedge \neg B) &\rightarrow B \vee C \vee \neg(A \wedge \neg C)\\ \neg(A \wedge \neg B) &\vee (B \vee C \vee \neg(A \wedge \neg C))\\ \neg A \vee \neg \neg B &\vee B \vee C \vee \neg(A \wedge \neg C)\\ \neg A \vee B &\vee B \vee C \vee \neg A \vee \neg \neg C\\ \neg A &\vee B \vee C\\ \end{aligned} $$
Here I've used DeMorgan's laws twice.