Key Concepts and Formulas

• Implication: $p \Rightarrow q$ is equivalent to $\sim p \lor q$.
• Equivalence: $p \Leftrightarrow q$ is equivalent to $(p \Rightarrow q) \land (q \Rightarrow p)$. Also, $p \Leftrightarrow q$ is true if and only if $p$ and $q$ have the same truth value.
• Tautology: A statement that is always true, regardless of the truth values of its components.

Step-by-Step Solution

Step 1: Analyze the given tautology

We are given that $(p \Rightarrow q) \Leftrightarrow (q * (\sim p))$ is a tautology. This means that the expressions $(p \Rightarrow q)$ and $(q * (\sim p))$ have the same truth values for all possible truth values of $p$ and $q$. Therefore, $p \Rightarrow q$ is equivalent to $q * (\sim p)$. We can write this as: $p \Rightarrow q \equiv q * (\sim p)$

Step 2: Rewrite the implication using disjunction

Recall that $p \Rightarrow q$ is equivalent to $\sim p \lor q$. Substituting this into the equivalence from Step 1, we get: $\sim p \lor q \equiv q * (\sim p)$

Step 3: Determine the operation represented by '*'

Since $\sim p \lor q \equiv q * (\sim p)$, we can deduce that the operation $*$ represents disjunction ($\lor$). Therefore, we can replace $*$ with $\lor$: $q * (\sim p) \equiv q \lor (\sim p)$

Step 4: Simplify the expression $p * (\sim q)$ using the identified operation

We want to find an equivalent expression for $p * (\sim q)$. Since we've determined that $*$ is equivalent to $\lor$, we have: $p * (\sim q) \equiv p \lor (\sim q)$

Step 5: Rewrite the disjunction as an implication

Recall that $\sim a \lor b$ is equivalent to $a \Rightarrow b$. Applying this to our expression, we have: $p \lor (\sim q) \equiv \sim(\sim p) \lor (\sim q) \equiv \sim q \lor p \equiv q \Rightarrow p$

Step 6: State the final equivalent expression

Therefore, $p * (\sim q)$ is equivalent to $q \Rightarrow p$.

Common Mistakes & Tips

• Remember the fundamental equivalences for implication and equivalence. A common mistake is to confuse $p \Rightarrow q$ with $q \Rightarrow p$.
• Recognize that if an equivalence is a tautology, the expressions on either side must have identical truth tables. This allows you to deduce the unknown operation.
• When simplifying Boolean expressions, it can be helpful to rewrite implications as disjunctions and vice versa.

Summary

We are given that $(p \Rightarrow q) \Leftrightarrow (q * (\sim p))$ is a tautology. By rewriting the implication as a disjunction and comparing the two sides of the equivalence, we determined that the operation $*$ represents disjunction ($\lor$). Then, we simplified the expression $p * (\sim q)$ as $p \lor (\sim q)$, which is equivalent to $q \Rightarrow p$.

Final Answer

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