Key Concepts and Formulas

• Implication (Conditional Statement): $p \Rightarrow q$ is logically equivalent to $\neg p \vee q$.
• Associativity of Conjunction: $(p \wedge q) \wedge r \equiv p \wedge (q \wedge r)$.
• Idempotent Law: $p \wedge p \equiv p$
• Commutativity of Conjunction: $p \wedge q \equiv q \wedge p$.
• Simplification: $(p \wedge q) \Rightarrow p$ is a tautology.

Step-by-Step Solution

Step 1: Rewrite the given expression using the definition of implication. We have $(p \wedge q) \Rightarrow ((r \wedge q) \wedge p)$. Using the implication rule $a \Rightarrow b \equiv \neg a \vee b$, we get $\neg(p \wedge q) \vee ((r \wedge q) \wedge p)$

Step 2: Apply De Morgan's Law. De Morgan's Law states that $\neg(p \wedge q) \equiv \neg p \vee \neg q$. Substituting this into the expression, we have $(\neg p \vee \neg q) \vee ((r \wedge q) \wedge p)$

Step 3: Rearrange and regroup terms using associativity and commutativity. Using the associative property of disjunction, we can rewrite the expression as $\neg p \vee \neg q \vee (r \wedge q \wedge p)$ Using the commutative property of conjunction, we can rewrite $r \wedge q \wedge p$ as $p \wedge q \wedge r$. $\neg p \vee \neg q \vee (p \wedge q \wedge r)$

Step 4: Regroup terms to isolate (p $\wedge$ q). We can rewrite the expression as $(\neg p \vee \neg q) \vee ((p \wedge q) \wedge r)$

Step 5: Apply the implication rule in reverse. Notice that $\neg p \vee \neg q$ is equivalent to $\neg (p \wedge q)$. Thus, we can rewrite the entire expression as: $\neg (p \wedge q) \vee ((p \wedge q) \wedge r)$ Now, we use the implication rule in reverse to get: $(p \wedge q) \Rightarrow ((p \wedge q) \wedge r)$

Step 6: Simplify the expression. The expression $(p \wedge q) \Rightarrow ((p \wedge q) \wedge r)$ can be simplified. We know that $a \Rightarrow (a \wedge b)$ is equivalent to $a \Rightarrow b$. Therefore, the expression becomes $(p \wedge q) \Rightarrow r$

Step 7: Manipulate the correct answer. We are given that the correct answer is $(p \wedge q) \Rightarrow (r \wedge q)$. This can be rewritten as $\neg (p \wedge q) \vee (r \wedge q)$ $(\neg p \vee \neg q) \vee (r \wedge q)$ $\neg p \vee \neg q \vee (r \wedge q)$ Since we have to prove $(p \wedge q) \Rightarrow r$ is equivalent to $(p \wedge q) \Rightarrow (r \wedge q)$, let's rewrite $(p \wedge q) \Rightarrow r$ as $\neg(p \wedge q) \vee r = (\neg p \vee \neg q) \vee r$. Now, the expression $(\neg p \vee \neg q) \vee r$ is equivalent to $(\neg p \vee \neg q) \vee (r \wedge (p \vee \neg p)) = (\neg p \vee \neg q) \vee ((r \wedge p) \vee (r \wedge \neg p))$. This doesn't seem to be simplifying to the correct answer.

Let's go back to $(p \wedge q) \Rightarrow ((r \wedge q) \wedge p)$. Since $(r \wedge q) \wedge p \equiv r \wedge q \wedge p \equiv r \wedge (q \wedge p) \equiv r \wedge (p \wedge q)$, we have $(p \wedge q) \Rightarrow (r \wedge (p \wedge q))$ This is equivalent to $\neg (p \wedge q) \vee (r \wedge (p \wedge q))$. The correct answer is $(p \wedge q) \Rightarrow (r \wedge q)$. This is equivalent to $\neg(p \wedge q) \vee (r \wedge q)$. We want to show that $\neg (p \wedge q) \vee (r \wedge (p \wedge q))$ is equivalent to $\neg(p \wedge q) \vee (r \wedge q)$.

Let $x = p \wedge q$. Then we have $\neg x \vee (r \wedge x)$ and we want to show it is equivalent to $\neg x \vee (r \wedge q)$. $\neg x \vee (r \wedge x) \equiv (\neg x \vee r) \wedge (\neg x \vee x) \equiv (\neg x \vee r) \wedge T \equiv \neg x \vee r$. $\neg(p \wedge q) \vee r = (p \wedge q) \Rightarrow r$ $\neg(p \wedge q) \vee (r \wedge q) = (p \wedge q) \Rightarrow (r \wedge q)$.

Now, we want to show $(p \wedge q) \Rightarrow ((r \wedge q) \wedge p)$ is equivalent to $(p \wedge q) \Rightarrow (r \wedge q)$. $(p \wedge q) \Rightarrow ((r \wedge q) \wedge p)$ simplifies to $(p \wedge q) \Rightarrow (r \wedge q \wedge p)$. Since we have $p \wedge q$ on the left, $r \wedge q \wedge p$ is equivalent to $r \wedge (p \wedge q)$. So we have $(p \wedge q) \Rightarrow (r \wedge (p \wedge q))$. Now, if $p \wedge q$ is true, then this simplifies to $r \wedge (p \wedge q)$. And $(p \wedge q) \Rightarrow (r \wedge q)$ simplifies to: if $p \wedge q$ is true, then $r \wedge q$ must be true. Since $q$ is true because $p \wedge q$ is true, then $r$ must be true. So $(p \wedge q) \Rightarrow r$ must be true.

Thus $(p \wedge q) \Rightarrow (r \wedge (p \wedge q))$ means that if $p \wedge q$ is true, then $r$ must also be true. That's $(p \wedge q) \Rightarrow r$. $(p \wedge q) \Rightarrow (r \wedge q)$ also means that if $p \wedge q$ is true, then $r \wedge q$ must be true. Since $q$ is true, $r$ must also be true. So we have $(p \wedge q) \Rightarrow r$.

Both are equivalent. So the answer is (A).

Common Mistakes & Tips

• Carefully apply De Morgan's Laws and the implication rule. A sign error can lead to a completely different result.
• Remember the associative and commutative properties to rearrange terms effectively.
• When simplifying, look for common sub-expressions and use substitution to make the process clearer.

Summary

We began by converting the implication to its equivalent disjunction form. We then applied De Morgan's laws and rearranged the terms to isolate common factors. We ultimately showed that the original expression is equivalent to $(p \wedge q) \Rightarrow (r \wedge q)$. This matches option (A).

Final Answer The final answer is \boxed{(p \wedge q) \Rightarrow (r \wedge q)}, which corresponds to option (A).