Key Concepts and Formulas

• Truth Tables: A table showing all possible combinations of truth values for statements and the resulting truth value of a compound statement.
• Logical Equivalences: Statements that have the same truth value for all possible truth values of their component propositions.
• Distributive Law: $p \vee (q \wedge r) \equiv (p \vee q) \wedge (p \vee r)$ and $p \wedge (q \vee r) \equiv (p \wedge q) \vee (p \wedge r)$
• Complement Law: $p \vee (\sim p) \equiv t$ (tautology) and $p \wedge (\sim p) \equiv f$ (contradiction)
• Identity Law: $p \wedge t \equiv p$ and $p \vee f \equiv p$
• DeMorgan's Laws: $\sim (p \wedge q) \equiv (\sim p) \vee (\sim q)$ and $\sim (p \vee q) \equiv (\sim p) \wedge (\sim q)$

Step-by-Step Solution

Step 1: Rewrite the given statement. WHY: Simply restating the given expression helps to keep track of what we are working with. $(p \wedge(\sim q)) \vee((\sim p) \wedge q) \vee((\sim p) \wedge(\sim q))$

Step 2: Combine the last two terms using the distributive law. WHY: Factoring out $(\sim p)$ simplifies the expression. $(p \wedge(\sim q)) \vee((\sim p) \wedge (q \vee (\sim q)))$

Step 3: Simplify the expression within the parenthesis $(q \vee (\sim q))$. WHY: Applying the complement law. $(p \wedge(\sim q)) \vee((\sim p) \wedge t)$ Since $q \vee (\sim q)$ is always true, we replace it with $t$.

Step 4: Simplify the expression $(\sim p) \wedge t$. WHY: Applying the identity law. $(p \wedge(\sim q)) \vee (\sim p)$

Step 5: Rearrange the terms. WHY: To prepare for applying the distributive law. $(\sim p) \vee (p \wedge (\sim q))$

Step 6: Apply the distributive law in reverse. WHY: To simplify the expression. $((\sim p) \vee p) \wedge ((\sim p) \vee (\sim q))$

Step 7: Simplify the expression $(\sim p) \vee p$. WHY: Applying the complement law. $t \wedge ((\sim p) \vee (\sim q))$ Since $(\sim p) \vee p$ is always true, we replace it with $t$.

Step 8: Simplify the expression $t \wedge ((\sim p) \vee (\sim q))$. WHY: Applying the identity law. $(\sim p) \vee (\sim q)$

Therefore, the given statement is equivalent to $(\sim p) \vee (\sim q)$.

Common Mistakes & Tips

• Incorrect application of distributive law: Ensure you apply the distributive law correctly, especially when "factoring out" terms.
• Misunderstanding logical equivalences: Familiarize yourself with common logical equivalences to simplify expressions efficiently.
• Using truth tables inefficiently: While truth tables can verify equivalences, algebraic simplification is often faster and more insightful.

Summary

We started with the given logical expression and systematically simplified it using logical equivalences such as the distributive law, complement law, and identity law. By combining terms, applying these laws, and rearranging the expression, we arrived at the simplified form $(\sim p) \vee (\sim q)$. This shows that the original statement is logically equivalent to $(\sim p) \vee (\sim q)$.

Final Answer

The final answer is \boxed{(\sim p) \vee (\sim q)}, which corresponds to option (A).