Key Concepts and Formulas

• Implication: $p \Rightarrow q \equiv \sim p \vee q$
• Tautology: A statement that is always true, regardless of the truth values of its components.
• Commutative Law: $p \vee q \equiv q \vee p$ and $p \wedge q \equiv q \wedge p$
• Associative Law: $(p \vee q) \vee r \equiv p \vee (q \vee r)$ and $(p \wedge q) \wedge r \equiv p \wedge (q \wedge r)$
• Law of Excluded Middle: $p \vee \sim p \equiv T$ (where T represents "True")
• Identity Law: $p \vee T \equiv T$ and $p \wedge F \equiv F$ (where F represents "False")

Step-by-Step Solution

Step 1: Analyze option (A): $(p \Rightarrow q) \vee (\sim q \Rightarrow p)$

• We want to simplify the expression using the implication rule.
• $(p \Rightarrow q) \vee (\sim q \Rightarrow p) \equiv (\sim p \vee q) \vee (\sim(\sim q) \vee p)$
• Simplifying, we get: $(\sim p \vee q) \vee (q \vee p)$

Step 2: Further simplification using commutative and associative laws

• Rearrange the terms using commutative and associative properties:
• $(\sim p \vee q) \vee (q \vee p) \equiv (\sim p \vee p) \vee (q \vee q)$
• Since $q \vee q \equiv q$, we have: $(\sim p \vee p) \vee q$

Step 3: Apply the Law of Excluded Middle

• $\sim p \vee p \equiv T$
• Therefore, we have: $T \vee q$

Step 4: Apply the Identity Law

• $T \vee q \equiv T$
• So, option (A) is a tautology.

Step 5: Analyze option (B): $(q \Rightarrow p) \vee (\sim q \Rightarrow p)$

• We want to simplify the expression using the implication rule.
• $(q \Rightarrow p) \vee (\sim q \Rightarrow p) \equiv (\sim q \vee p) \vee (\sim(\sim q) \vee p)$
• Simplifying, we get: $(\sim q \vee p) \vee (q \vee p)$

Step 6: Further simplification using commutative and associative laws

• Rearrange the terms using commutative and associative properties:
• $(\sim q \vee p) \vee (q \vee p) \equiv (\sim q \vee q) \vee (p \vee p)$

Step 7: Apply the Law of Excluded Middle

• $\sim q \vee q \equiv T$
• Also, $p \vee p \equiv p$
• Therefore, we have: $T \vee p$

Step 8: Apply the Identity Law

• $T \vee p \equiv T$
• So, option (B) is a tautology.

Step 9: Analyze option (C): $(p \Rightarrow \sim q) \vee (\sim q \Rightarrow p)$

• We want to simplify the expression using the implication rule.
• $(p \Rightarrow \sim q) \vee (\sim q \Rightarrow p) \equiv (\sim p \vee \sim q) \vee (\sim(\sim q) \vee p)$
• Simplifying, we get: $(\sim p \vee \sim q) \vee (q \vee p)$

Step 10: Further simplification using commutative and associative laws

• Rearrange the terms using commutative and associative properties:
• $(\sim p \vee \sim q) \vee (q \vee p) \equiv (\sim p \vee p) \vee (\sim q \vee q)$

Step 11: Apply the Law of Excluded Middle

• $\sim p \vee p \equiv T$ and $\sim q \vee q \equiv T$
• Therefore, we have: $T \vee T$

Step 12: Apply the Identity Law

• $T \vee T \equiv T$
• So, option (C) is a tautology.

Step 13: Analyze option (D): $(\sim p \Rightarrow q) \vee (\sim q \Rightarrow p)$

• We want to simplify the expression using the implication rule.
• $(\sim p \Rightarrow q) \vee (\sim q \Rightarrow p) \equiv (\sim(\sim p) \vee q) \vee (\sim(\sim q) \vee p)$
• Simplifying, we get: $(p \vee q) \vee (q \vee p)$

Step 14: Further simplification using commutative and associative laws

• Rearrange the terms using commutative and associative properties:
• $(p \vee q) \vee (q \vee p) \equiv p \vee q \vee q \vee p \equiv p \vee q$

Step 15: Analyze the result.

• The expression $p \vee q$ is not a tautology. It is only true if either $p$ is true or $q$ is true, or both are true. If both $p$ and $q$ are false, then $p \vee q$ is false. Therefore, option (D) is not a tautology.

Common Mistakes & Tips

• Remember the correct definition of implication: $p \Rightarrow q \equiv \sim p \vee q$. A common mistake is to mix it up with other logical equivalences.
• When simplifying complex Boolean expressions, carefully apply commutative and associative laws to group terms that can be simplified using the Law of Excluded Middle.
• A tautology must be true for all possible truth values of its components. If you can find even one case where the expression is false, it's not a tautology.

Summary

We analyzed each of the given Boolean expressions to determine which one is not a tautology. By applying the implication rule, commutative and associative laws, the Law of Excluded Middle, and the Identity Law, we simplified each expression. Options (A), (B), and (C) simplified to $T$, indicating that they are tautologies. Option (D) simplified to $p \vee q$, which is not a tautology because it is false when both p and q are false.

Final Answer

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