Verified Solution • 2025–26 Session moderate

QJEEM23Ma5178: JEE Main 2023 Mathematics - Class 11 JEEM23Ma5178 Solution: The statement is equivalent to _________.

The statement (p∧(∼q))∨((∼p)∧q)∨((∼p)∧(∼q))(p \wedge(\sim q)) \vee((\sim p) \wedge q) \vee((\sim p) \wedge(\sim q)) is equivalent to _________.

Multiple Choice Options

  • A (∼p)∨(∼q)(\sim p) \vee(\sim q)
  • B p∨(∼q)p \vee(\sim q)
  • C p∨q\mathrm{p} \vee \mathrm{q}
  • D (∼p)∨q(\sim p) \vee q

Step-by-Step Verified Solution

pamp;qamp;∼qamp;∼pamp;p∧∼qamp;∼p∧qamp;∼p∧∼qTamp;Tamp;Famp;Famp;Famp;Famp;FTamp;Famp;Tamp;Famp;Tamp;Famp;FFamp;Tamp;Famp;Tamp;Famp;Tamp;FFamp;Famp;Tamp;Tamp;Famp;Famp;T\begin{array}{|c|c|c|c|c|c|c|}\hline \mathbf{p} & \mathbf{q} & \sim \mathbf{q} & \sim \mathbf{p} & \mathbf{p} \wedge \sim \mathbf{q} & \mathbf{\sim p} \wedge \mathbf{q} & \mathbf{\sim p} \wedge \mathbf{\sim q} \\\hline T & T & F & F & F & F & F \\T & F & T & F & T & F & F \\F & T & F & T & F & T & F \\F & F & T & T & F & F & T \\\hline\end{array}

Using the truth table, we can see that the given statement is equivalent to (∼p)∨(∼q)(\sim p) \vee(\sim q) .

Alternate Method :

The given statement (p∧(∼q))∨((∼p)∧q)∨((∼p)∧(∼q))(p \wedge(\sim q)) \vee((\sim p) \wedge q) \vee((\sim p) \wedge(\sim q)) can be simplified using the following steps:

amp;(p∧(∼q))∨((∼p)∧q)∨((∼p)∧(∼q))amp;=(p∧(∼q))∨((∼p)∧(q∨(∼q)))amp;=(p∧(∼q))∨((∼p)∧t)amp;=(p∧(∼q))∨(∼p)amp;=(∼p)∨(p∧(∼q))amp;=(∼p∨p)∧(∼p∨∼q)amp;=t∧(∼p∨∼q)amp;=∼p∨∼q\begin{aligned}&(p \wedge(\sim q)) \vee((\sim p) \wedge q) \vee((\sim p) \wedge(\sim q)) \\\\&= \left(p \wedge(\sim q)\right) \vee \left((\sim p) \wedge (q \vee (\sim q))\right) \\\\&= \left(p \wedge(\sim q)\right) \vee \left((\sim p) \wedge t\right) \\\\&= \left(p \wedge(\sim q)\right) \vee (\sim p) \\\\&= (\sim p) \vee \left(p \wedge (\sim q)\right) \\\\&= (\sim p \vee p) \wedge (\sim p \vee \sim q) \\\\&= t \wedge (\sim p \vee \sim q) \\\\&= \sim p \vee \sim q\end{aligned}

Therefore, the given statement is equivalent to (∼p)∨(∼q)(\sim p) \vee (\sim q) .

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