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JEE Main 20236 AprilEvening Shift
Exam
JEE Main
Year
2023
Date
6 April
Shift
Evening Shift
Subject
Mathematics
Chapter
Mathematical Reasoning

JEE Main 2023 · 6 April · Evening Shift · Mathematics · Mathematical Reasoning

Among the statements (S1): (p ⇒ q) ∨ ((~ p) ∧ q) is a tautology...

Among the statements

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

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

Options

  1. ANeither (S1)\left(S1\right) and (S2)\left(S2\right) is True
  2. BBoth (S1)\left(S1\right) and (S2)\left(S2\right) are True
  3. COnly (S2)\left(S2\right) is True
  4. DOnly (S1)\left(S1\right) is True

Correct answer

A. Neither (S1)\left(S1\right) and (S2)\left(S2\right) is True

Step-by-step solution

Plotting the Truth table of the given expression we get,

pp qq  p~ p  p∧q~ p ∧ q p⇒qp ⇒ q q⇒pq ⇒ p \left(p ⇒ q\right) ∨ \left(~ p ∧ q\right) \left(q ⇒ p\right) ∨ \left(~ p ∧ q\right)
T</td><td>T</td><td>F</td><td>F</td><td>T</td><td>T</td><td>T</td><td>T</td></tr><tr><td>T</td><td>F</td><td>F</td><td>F</td><td>F</td><td>T</td><td>F</td><td>T</td></tr><tr><td>F</td><td>T</td><td>T</td><td>T</td><td>T</td><td>F</td><td>T</td><td>T</td></tr><tr><td>F</td><td>F</td><td>T</td><td>F</td><td>T</td><td>T</td><td>T</td><td>T</td></tr></tbody></table><p>Hence,</td> <td>T</td> <td>F</td> <td>F</td> <td>T</td> <td>T</td> <td>T</td> <td>T</td> </tr> <tr> <td>T</td> <td>F</td> <td>F</td> <td>F</td> <td>F</td> <td>T</td> <td>F</td> <td>T</td> </tr> <tr> <td>F</td> <td>T</td> <td>T</td> <td>T</td> <td>T</td> <td>F</td> <td>T</td> <td>T</td> </tr> <tr> <td>F</td> <td>F</td> <td>T</td> <td>F</td> <td>T</td> <td>T</td> <td>T</td> <td>T</td> </tr> </tbody></table><p>Hence,\left(q ⇒ p\right)∨( p∧q)∨ \left(~ p ∧ q\right) is a tautology, and \left(p ⇒ q\right)∨( p∧q)∨ \left(~ p ∧ q\right) is neither tautology and contradiction,</p><p>Sobothstatementsarefalse.</p></blockquote></p><p>So both statements are false.</p></blockquote> Practice Mathematical Reasoning on Quantrex Academy →

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