1. **Problem:** Construct a detailed truth table for the formula $ (p \wedge q) \to (r \wedge s) $ and analyze whether it is a tautology. 2. **Step 1:** List all possible truth values for $p$, $q$, $r$, and $s$. There are $2^4 = 16$ combinations. 3. **Step 2:** Compute $p \wedge q$ and $r \wedge s$ for each combination. 4. **Step 3:** Compute the implication $ (p \wedge q) \to (r \wedge s) $. Recall that $A \to B$ is false only when $A$ is true and $B$ is false; otherwise true. 5. **Step 4:** Check if the formula is true in all cases. If yes, it is a tautology; if not, it is not. 6. **Summary:** Since $ (p \wedge q) \to (r \wedge s) $ is not always true (for example, when $p=q=\text{true}$ and $r=s=\text{false}$, the implication is false), it is not a tautology. --- 1. **Problem:** Describe the indirect method in propositional calculus and prove a step using an example. 2. **Step 1:** The indirect method (proof by contradiction) assumes the negation of what you want to prove. 3. **Step 2:** From this assumption, derive a contradiction (a statement and its negation). 4. **Step 3:** Conclude that the assumption is false, so the original statement is true. 5. **Example:** To prove $p \to q$, assume $p$ and $\neg q$. 6. **Step 4:** If this leads to a contradiction, then $p \to q$ holds. --- 1. **Problem:** Show that $N(p \wedge q \vee .)$ and $(p \wedge q \wedge)$ write qualified statement of systematic form and symbolic form $pn$. 2. **Step 1:** The notation is unclear; assuming $N$ is negation, and $\vee$ and $\wedge$ are OR and AND. 3. **Step 2:** For example, $N(p \wedge q \vee r)$ means $\neg((p \wedge q) \vee r)$. 4. **Step 3:** Systematic form: "It is not the case that either both $p$ and $q$ are true or $r$ is true." 5. **Step 4:** Symbolic form: $\neg((p \wedge q) \vee r)$. --- 1. **Problem:** Write qualified statement of systematic form and symbolic form for $\&n$ (assuming $\&n$ means conjunction with negation). 2. **Step 1:** For example, "Both $p$ and not $q$ are true." 3. **Step 2:** Symbolic form: $p \wedge \neg q$. --- 1. **Problem:** State the duality laws (important logical equations). 2. **Step 1:** Duality laws state that every algebraic expression remains valid if we interchange $\wedge$ and $\vee$ and interchange $\top$ (true) and $\bot$ (false). 3. **Step 2:** Examples: $$ (p \wedge q)^d = p \vee q $$ $$ (p \vee q)^d = p \wedge q $$ 4. **Step 3:** This helps in deriving dual statements and simplifying logical expressions. **Final answers:** - The formula $ (p \wedge q) \to (r \wedge s) $ is not a tautology. - The indirect method uses proof by contradiction. - Negation of conjunction or disjunction can be expressed systematically and symbolically. - Duality laws interchange $\wedge$ and $\vee$ and $\top$ and $\bot$.