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NTA Abhyas 2020
Exam
NTA Abhyas
Year
2020
Subject
Mathematics
Chapter
Mathematical Reasoning

NTA Abhyas 2020 · Mathematics · Mathematical Reasoning

If p and q are two statements, then which of the following...

If pandp andqaretwostatements,thenwhichofthefollowingstatementsisnotequivalenttoare two statements, then which of the following statements is not equivalent top ⇔ (p⇒q)?\left(p ⇒ q\right)?

Options

  1. Ap∧qp ∧ q
  2. B(p⇔q)∧(p∨q)\left(p ⇔ q\right) ∧ \left(p ∨ q\right)
  3. C(q⇒p)⇔q\left(q ⇒ p\right) ⇔ q
  4. D(q⇒p)∧(p⇒q)\left( q ⇒ p\right) ∧ \left(p ⇒ q\right)

Correct answer

D. (q⇒p)∧(p⇒q)\left( q ⇒ p\right) ∧ \left(p ⇒ q\right)

Step-by-step solution

pp qq p⇒qp ⇒ q p⇔(p⇒q)</th></tr></thead><tbody><tr><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></tr><tr><td>F</td><td>T</td><td>T</td><td>F</td></tr><tr><td>F</td><td>F</td><td>T</td><td>F</td></tr></tbody></table><br/>p ⇔ \left(p ⇒ q\right)</th> </tr> </thead> <tbody> <tr> <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> </tr> <tr> <td>F</td> <td>T</td> <td>T</td> <td>F</td> </tr> <tr> <td>F</td> <td>F</td> <td>T</td> <td>F</td> </tr> </tbody></table><br />⇒ p ⇔ \left(p ⇒ q\right) ≡ \left(p ∧ q\right) ⇒ \left(A\right) is true
Now, q⇔(q⇒p)q ⇔ \left(q ⇒ p\right) ≡ \left(p ∧ q\right)≡p∧q≡ p ∧ q
⇒(q⇒p)⇒ \left(q ⇒ p\right) ⇒ q ≡ p ∧ q ⇒ \left(C\right)istrue<br/>Now,commonbetween<br/>(p⇔q)is true<br />Now, common between<br /> \left(p ⇔ q\right) & \left(p ∨ q\right)is<br/>is<br />p ∧ q ⇒ \left(p ⇔ q\right) ∧ \left(p ∨ q\right) ≡ p ∧ q<br/><br />⇒ \left(B\right) is correct
Now, \left( q ⇒ p\right)∧ \left(p ⇒ q\right)≡ \left(q ∨ p\right)∧ \left( p ∨ q\right)
= Only pp or only q≡(p⇔q)q ≡ \left(p ⇔ q\right) Practice Mathematical Reasoning on Quantrex Academy →

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