Step 1: Understanding the Concept:
We need to simplify a logical expression using Boolean algebra laws or logical equivalences such as Distributive Law, Complement Law, and Identity Law.
Step 2: Key Formula or Approach:
Distributive Law: $A \land (B \lor C) \equiv (A \land B) \lor (A \land C)$.
Distributive Law: $A \lor (B \land C) \equiv (A \lor B) \land (A \lor C)$.
Complement Law: $A \lor \sim A \equiv T$.
Identity Law: $A \land T \equiv A$.
Step 3: Detailed Explanation:
Let the given expression be $S = (\sim p \land q) \lor (\sim p \land \sim q) \lor (p \land \sim q)$.
We can group the first two terms together:
$S = \left[ (\sim p \land q) \lor (\sim p \land \sim q) \right] \lor (p \land \sim q)$
Apply the Distributive Law in reverse (factoring out $\sim p$) to the grouped terms:
$\left[ (\sim p \land q) \lor (\sim p \land \sim q) \right] \equiv \sim p \land (q \lor \sim q)$
By the Complement Law, $q \lor \sim q \equiv T$ (True).
So, the grouped term simplifies to $\sim p \land T \equiv \sim p$.
Now substitute this back into $S$:
$S = \sim p \lor (p \land \sim q)$
Apply the Distributive Law to this new expression:
$S \equiv (\sim p \lor p) \land (\sim p \lor \sim q)$
Again, using the Complement Law, $\sim p \lor p \equiv T$.
$S \equiv T \land (\sim p \lor \sim q)$
By Identity Law, this simplifies to:
$S \equiv \sim p \lor \sim q$
Step 4: Final Answer:
The logically equivalent statement is $(\sim p) \lor (\sim q)$.