Key Concepts and Formulas

• Implication: $p \Rightarrow q \equiv \sim p \vee q$
• De Morgan's Laws: $\sim (p \wedge q) \equiv \sim p \vee \sim q$ and $\sim (p \vee q) \equiv \sim p \wedge \sim q$
• Distributive Laws: $p \vee (q \wedge r) \equiv (p \vee q) \wedge (p \vee r)$ and $p \wedge (q \vee r) \equiv (p \wedge q) \vee (p \wedge r)$
• Associative Laws: $(p \vee q) \vee r \equiv p \vee (q \vee r)$ and $(p \wedge q) \wedge r \equiv p \wedge (q \wedge r)$

Step-by-Step Solution

Let's analyze each option and transform it to see if it's equivalent to $(p \Rightarrow q) \vee (p \Rightarrow r)$, which is equivalent to $(\sim p \vee q) \vee (\sim p \vee r) \equiv \sim p \vee q \vee \sim p \vee r \equiv \sim p \vee q \vee r \equiv p \Rightarrow (q \vee r)$.

Step 1: Analyze Option (A): $(p \wedge (\sim r)) \Rightarrow q$

We start by converting the implication to its equivalent form using the formula $p \Rightarrow q \equiv \sim p \vee q$. $(p \wedge (\sim r)) \Rightarrow q \equiv \sim(p \wedge \sim r) \vee q$

Step 2: Apply De Morgan's Law

Apply De Morgan's Law to simplify $\sim(p \wedge \sim r)$. $\sim(p \wedge \sim r) \vee q \equiv (\sim p \vee \sim(\sim r)) \vee q \equiv (\sim p \vee r) \vee q$

Step 3: Apply Associative Law

Use the associative law to rearrange the terms. $(\sim p \vee r) \vee q \equiv \sim p \vee (r \vee q) \equiv \sim p \vee (q \vee r)$

Step 4: Convert back to Implication

Convert the expression back to implication form. $\sim p \vee (q \vee r) \equiv p \Rightarrow (q \vee r)$

Step 5: Compare with the original statement

We know that $p \Rightarrow (q \vee r) \equiv (\sim p \vee q) \vee (\sim p \vee r) \equiv (p \Rightarrow q) \vee (p \Rightarrow r)$. Therefore, option (A) is equivalent to the original statement.

Step 6: Analyze Option (B): $(\sim q) \Rightarrow ((\sim r) \vee p)$

Convert the implication to its equivalent form. $(\sim q) \Rightarrow (\sim r \vee p) \equiv \sim (\sim q) \vee (\sim r \vee p) \equiv q \vee \sim r \vee p \equiv \sim r \vee (p \vee q)$

Step 7: Re-arrange the terms $\sim r \vee (p \vee q) \equiv p \vee (q \vee \sim r) \equiv \sim p \Rightarrow (q \vee \sim r)$ The expression is NOT equivalent to $p \Rightarrow (q \vee r)$. Let's manipulate the original expression. $(\sim q) \Rightarrow ((\sim r) \vee p) \equiv (\sim q) \Rightarrow (p \vee (\sim r)) \equiv q \vee (p \vee (\sim r)) \equiv p \vee q \vee (\sim r) \equiv \sim p \Rightarrow (q \vee \sim r)$ Now we want to show that it is NOT equivalent to $p \Rightarrow (q \vee r) \equiv \sim p \vee q \vee r$. For the two to be equivalent, we need: $p \vee q \vee (\sim r) \equiv \sim p \vee q \vee r$ $p \vee (\sim r) \equiv \sim p \vee r$ However, option B is equivalent to option C.

Step 8: Analyze Option (C): $p \Rightarrow (q \vee r)$

Convert the implication to its equivalent form. $p \Rightarrow (q \vee r) \equiv \sim p \vee (q \vee r)$

Step 9: Apply Associative Law

Apply the associative law. $\sim p \vee (q \vee r) \equiv (\sim p \vee q) \vee r \equiv (\sim p \vee q) \vee (\sim p \vee r) \equiv (p \Rightarrow q) \vee (p \Rightarrow r)$ Therefore, option (C) is equivalent to the original statement.

Step 10: Analyze Option (D): $(p \wedge (\sim q)) \Rightarrow r$

Convert the implication to its equivalent form. $(p \wedge (\sim q)) \Rightarrow r \equiv \sim(p \wedge \sim q) \vee r$

Step 11: Apply De Morgan's Law

Apply De Morgan's Law to simplify $\sim(p \wedge \sim q)$. $\sim(p \wedge \sim q) \vee r \equiv (\sim p \vee \sim(\sim q)) \vee r \equiv (\sim p \vee q) \vee r$

Step 12: Apply Associative Law

Use the associative law to rearrange the terms. $(\sim p \vee q) \vee r \equiv \sim p \vee (q \vee r)$

Step 13: Convert back to Implication

Convert the expression back to implication form. $\sim p \vee (q \vee r) \equiv p \Rightarrow (q \vee r)$

Step 14: Compare with the original statement

We know that $p \Rightarrow (q \vee r) \equiv (\sim p \vee q) \vee (\sim p \vee r) \equiv (p \Rightarrow q) \vee (p \Rightarrow r)$. Therefore, option (D) is equivalent to the original statement.

Step 15: Revisit Option (B) We made an error in Step 7. Let's re-evaluate Option (B): $(\sim q) \Rightarrow ((\sim r) \vee p)$. $(\sim q) \Rightarrow ((\sim r) \vee p) \equiv q \vee (\sim r \vee p) \equiv q \vee \sim r \vee p \equiv p \vee q \vee \sim r$. The original statement is $p \Rightarrow (q \vee r) \equiv \sim p \vee q \vee r$. Thus, we want to see if $p \vee q \vee \sim r \equiv \sim p \vee q \vee r$. This is not always the case. Let $p=F, q=F, r=F$. Then $F \vee F \vee T \equiv T$ and $T \vee F \vee F \equiv T$. Let $p=T, q=F, r=F$. Then $T \vee F \vee T \equiv T$ and $F \vee F \vee F \equiv F$. So Option B is NOT equivalent.

Common Mistakes & Tips

• Be careful when applying De Morgan's Laws; double negations are a frequent source of errors.
• Remember the equivalence between implication and disjunction: $p \Rightarrow q \equiv \sim p \vee q$.
• When comparing logical statements, try to reduce them to their simplest forms.

Summary

We analyzed each option and transformed them to see if they are equivalent to the original statement $(p \Rightarrow q) \vee (p \Rightarrow r)$. Options (A), (C), and (D) were found to be equivalent, while Option (B) is not.

Final Answer

The final answer is \boxed{B}, which corresponds to option (B).