Key Concepts and Formulas

• Tautology: A statement that is always true, regardless of the truth values of its components.
• Implication ($p \Rightarrow q$): This is false only when $p$ is true and $q$ is false. Otherwise, it is true. $p \Rightarrow q$ is equivalent to $\sim p \vee q$.
• Negation ($\sim p$): If $p$ is true, then $\sim p$ is false, and vice versa.
• Disjunction ($p \vee q$): This is true if at least one of $p$ or $q$ is true. It is false only when both $p$ and $q$ are false.

Step-by-Step Solution

We will determine which of the given options is a tautology by analyzing their truth tables or using logical equivalences.

Option (A): $((\sim p) \vee q) \Rightarrow p$

Step 1: Rewrite the implication using the equivalence $a \Rightarrow b \equiv \sim a \vee b$. $((\sim p) \vee q) \Rightarrow p \equiv \sim((\sim p) \vee q) \vee p$

Step 2: Apply De Morgan's Law: $\sim(a \vee b) \equiv (\sim a \wedge \sim b)$. $\sim((\sim p) \vee q) \vee p \equiv (\sim(\sim p) \wedge \sim q) \vee p$

Step 3: Simplify $\sim(\sim p)$ to $p$. $(p \wedge \sim q) \vee p$

Step 4: Apply the absorption law: $(a \wedge b) \vee a \equiv a$. $(p \wedge \sim q) \vee p \equiv p$ Since this is just $p$, it is not a tautology as its truth value depends on the truth value of $p$.

Option (B): $p \Rightarrow((\sim p) \vee q)$

Step 1: Rewrite the implication using the equivalence $a \Rightarrow b \equiv \sim a \vee b$. $p \Rightarrow((\sim p) \vee q) \equiv \sim p \vee ((\sim p) \vee q)$

Step 2: Since disjunction is associative, we can rewrite this as: $\sim p \vee (\sim p \vee q) \equiv (\sim p \vee \sim p) \vee q$

Step 3: Simplify $\sim p \vee \sim p$ to $\sim p$. $(\sim p \vee \sim p) \vee q \equiv \sim p \vee q$ This is not a tautology since it is false when $p$ is true and $q$ is false.

Option (C): $((\sim p) \vee q) \Rightarrow q$

Step 1: Rewrite the implication using the equivalence $a \Rightarrow b \equiv \sim a \vee b$. $((\sim p) \vee q) \Rightarrow q \equiv \sim((\sim p) \vee q) \vee q$

Step 2: Apply De Morgan's Law: $\sim(a \vee b) \equiv (\sim a \wedge \sim b)$. $\sim((\sim p) \vee q) \vee q \equiv (\sim(\sim p) \wedge \sim q) \vee q$

Step 3: Simplify $\sim(\sim p)$ to $p$. $(p \wedge \sim q) \vee q$

Step 4: Distribute the disjunction. $(p \wedge \sim q) \vee q \equiv (p \vee q) \wedge (\sim q \vee q)$

Step 5: Since $\sim q \vee q$ is always true, it simplifies to $T$ (True). $(p \vee q) \wedge T \equiv p \vee q$ This is not a tautology since it is false when $p$ is false and $q$ is false.

Option (D): $q \Rightarrow((\sim p) \vee q)$

Step 1: Rewrite the implication using the equivalence $a \Rightarrow b \equiv \sim a \vee b$. $q \Rightarrow((\sim p) \vee q) \equiv \sim q \vee ((\sim p) \vee q)$

Step 2: Since disjunction is associative, we can rewrite this as: $\sim q \vee ((\sim p) \vee q) \equiv \sim q \vee (q \vee \sim p)$

Step 3: Since disjunction is commutative, we can rewrite this as: $\sim q \vee (q \vee \sim p) \equiv (\sim q \vee q) \vee \sim p$

Step 4: Simplify $\sim q \vee q$ to $T$ (True). $(\sim q \vee q) \vee \sim p \equiv T \vee \sim p$

Step 5: Since $T \vee a$ is always true, the entire expression is a tautology. $T \vee \sim p \equiv T$ This is a tautology. Therefore, option (D) is the answer.

There seems to be an error in the "Correct Answer" provided. The correct answer according to our derivation is (D), not (A).

Common Mistakes & Tips

• Remember De Morgan's Laws and how to negate compound statements.
• Be careful with the order of operations when constructing truth tables.
• Use logical equivalences to simplify expressions before constructing truth tables, which can save time.

Summary

We analyzed each of the given options to determine which one represents a tautology. By using logical equivalences and truth table principles, we found that option (D), $q \Rightarrow((\sim p) \vee q)$, is a tautology because it simplifies to $T$ (True) regardless of the truth values of $p$ and $q$.

Final Answer

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