Verified Solution • 2025–26 Session moderate

QJEEM23Ma8eac: JEE Main 2023 Mathematics - Class 11 JEEM23Ma8eac Solution: Which of the following statements is a tautology?

Which of the following statements is a tautology?

Multiple Choice Options

  • A p∨(p∧q)\mathrm{p\vee(p\wedge q)}
  • B (p∧(p→q))→ ∼q(\mathrm{p\wedge(p\to q))\to\,\sim q}
  • C p→(p∧(p→q))\mathrm{p\to (p\wedge (p\to q))}
  • D (p∧q)→(∼(p)→q)(\mathrm{p\wedge q)\to(\sim (p)\to q)}

Step-by-Step Verified Solution

amp;∼p→q≡∼(∼p)∨q≡p∨qamp;p∧q→(∼p→q)amp;≡p∧q→(p∨q)amp;≡∼(p∧q)∨(p∨q)amp;≡(∼p∨∼q)∨(p∨q)amp;≡(∼p∨(p∨q))∨(∼q∨(p∨q))amp;≡T∨Tamp;≡T\begin{aligned} & \sim p \rightarrow q \equiv \sim(\sim p) \vee q \equiv p \vee q \\\\ & p \wedge q \rightarrow(\sim p \rightarrow q) \\\\ & \equiv p \wedge q \rightarrow(p \vee q) \\\\ & \equiv \sim(p \wedge q) \vee(p \vee q) \\\\ & \equiv(\sim p \vee \sim q) \vee(p \vee q) \\\\ & \equiv(\sim p \vee(p \vee q)) \vee(\sim q \vee(p \vee q)) \\\\ & \equiv T \vee T \\\\ & \equiv T\end{aligned}

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