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JEE Main 202331 JanuaryMorning Shift
Exam
JEE Main
Year
2023
Date
31 January
Shift
Morning Shift
Subject
Mathematics
Chapter
Mathematical Reasoning

JEE Main 2023 · 31 January · Morning Shift · Mathematics · Mathematical Reasoning

Consider: (S1): (p ⇒ q) ∨ (p ∧ (~ q)) is a tautology. (S2): ((~ p)...

Consider:

(S1):(p⇒q)∨(p∧( q))\left(S1\right): \left(p ⇒ q\right) ∨ \left(p ∧ \left(~ q\right)\right) is a tautology.

\left(S2\right): (( p)⇒( q))∧(( p)∨q)\left(\left(~ p\right) ⇒ \left(~ q\right)\right) ∧ \left(\left(~ p\right) ∨ q\right) is a contradiction.

Then

Options

  1. Aonly (S2)\left(S2\right) is correct
  2. Bboth (S1)\left(S1\right) and (S2)\left(S2\right) are correct
  3. Cboth (S1)\left(S1\right) and (S2)\left(S2\right) are wrong
  4. Donly (S1)\left(S1\right) is correct

Correct answer

D. only (S1)\left(S1\right) is correct

Step-by-step solution

pp qq p⇒qp ⇒ q  q~ q p∧ qp ∧ ~ q \left(p ⇒ q\right) ∨ \left(p ∧ ~ q\right)</td></tr><tr><td></td> </tr> <tr> <td>T</td><td></td> <td>T</td><td></td> <td>T</td><td></td> <td>F</td><td></td> <td>F</td><td></td> <td>T</td></tr><tr><td></td> </tr> <tr> <td>T</td><td></td> <td>F</td><td></td> <td>F</td><td></td> <td>T</td><td></td> <td>T</td><td></td> <td>T</td></tr><tr><td></td> </tr> <tr> <td>F</td><td></td> <td>T</td><td></td> <td>T</td><td></td> <td>F</td><td></td> <td>F</td><td></td> <td>T</td></tr><tr><td></td> </tr> <tr> <td>F</td><td></td> <td>F</td><td></td> <td>T</td><td></td> <td>T</td><td></td> <td>F</td><td></td> <td>T</td></tr></tbody></table><p>So,</td> </tr> </tbody></table><p>So,\left(p ⇒ q\right) ∨ (p∧ q)\left(p ∧ ~ q\right) is a tautology.

 p~ p  q~ q  p⇒ q~ p ⇒ ~ q  p∨q~ p ∨ q \left(\left(~ p\right) ⇒ \left(~ q\right)\right) ∧ \left(\left(~ p\right) ∨ q\right)</td></tr><tr><td></td> </tr> <tr> <td>F</td><td></td> <td>F</td><td></td> <td>T</td><td></td> <td>T</td><td></td> <td>T</td></tr><tr><td></td> </tr> <tr> <td>F</td><td></td> <td>T</td><td></td> <td>T</td><td></td> <td>F</td><td></td> <td>F</td></tr><tr><td></td> </tr> <tr> <td>T</td><td></td> <td>F</td><td></td> <td>F</td><td></td> <td>T</td><td></td> <td>F</td></tr><tr><td></td> </tr> <tr> <td>T</td><td></td> <td>T</td><td></td> <td>T</td><td></td> <td>T</td><td></td> <td>T
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