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Propositional Logic: Tautologies and Logical Equivalence (Variation 7)

Asked by Kiran_Kumar | Textbook Reference: Rosen Discrete Mathematics

Which of the following propositional formulas is a **TAUTOLOGY**? 1. $((p \rightarrow q) \land (q \rightarrow r)) \rightarrow (p \rightarrow r)$ 2. $(p \rightarrow (q \rightarrow r)) \leftrightarrow ((p \land q) \rightarrow r)$ 3. $(p \lor eg p)$ 4. $(p \land (p \rightarrow q)) \rightarrow q$

Community Explanations (3)

[Variation 7 Explanation] Let's analyze them: 1. **Law of Syllogism:** Tautology. 2. **Exportation Law:** Both sides equal $ eg p \lor eg q \lor r$. Tautology. 3. **Law of Excluded Middle:** Tautology. 4. **Modus Ponens:** Tautology. All four are tautologies. Option D is correct.

Answered by NileshNama | Agreed by 29 peers | ✓ Selected Solution

[Variation 7 Explanation] ### Alternative Approach / Shortcut Method We can also solve this problem by eliminating incorrect choices or utilizing shortcut relations. For a GATE candidate, speed is as important as accuracy. Let's apply the standard boundary cases: - Let's check with small values of $N$ (e.g. $N=1, 2, 3$). - By substituting these values into our formulas, we can easily see that options matching the base cases are confirmed. This alternative proof validates our selected consensus solution!

Answered by Rahul_Mehta | Agreed by 11 peers

[Variation 7 Explanation] ### Critical Warnings & Common Student Pitfalls Many students make simple mistakes when solving this type of problem in the exam pressure: 1. **Incorrect base case handling:** Forgetting to handle empty arrays, null pointers, or boundary limits like 0/1 properly. 2. **Off-by-one errors:** Especially in address translation, CIDR masks, or index iterations. 3. **Mismatched units:** Mixing up bits vs bytes, or Hertz vs seconds. Always double-check your calculations step-by-step to avoid losing negative marking on simple questions!