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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 logical statements, then p ⇒ ( q ⇒ p) is equivalent...

If pp and qq are logical statements, then p⇒(q⇒p)isequivalenttop ⇒ \left( q ⇒ p\right) is equivalent to

Options

  1. Ap⇒(p⇒q)p ⇒ \left(p ⇒ q\right)
  2. Bp⇒(p∨q)p ⇒ \left(p ∨ q\right)
  3. Cp⇒(p∧q)p ⇒ \left(p ∧ q\right)
  4. Dp⇒(p⇔q)p ⇒ \left(p ⇔ q\right)

Correct answer

B. p⇒(p∨q)p ⇒ \left(p ∨ q\right)

Step-by-step solution

p⇒(q⇒p)≡∼p∨(q∨p)p ⇒ \left( q ⇒ p\right) ≡ ∼ p ∨ \left(q ∨ p\right)
≡(∼p∨p)∨(q)≡t∨(q)≡t≡ \left(∼ p ∨ p\right) ∨ \left(q\right) ≡ t ∨ \left(q\right) ≡ t
Hence, the given statement p⇒(q⇒p)p ⇒ \left( q ⇒ p\right) is a tautology
Now, p⇒(p⇒q)p ⇒ \left(p ⇒ q\right) is false when pp is true and qq is false

Hence, not a tautology
p⇒(q∨p)≡p∨(p∨q)≡(p∨p)∨qp ⇒ \left(q ∨ p\right) ≡ p ∨ \left(p ∨ q\right) ≡ \left( p ∨ p\right) ∨ q
t∨q≡tt ∨ q ≡ t
Hence, p⇒(q∨p)p ⇒ \left(q ∨ p\right) is a tautology.
Now, p⇒(p∧q)p ⇒ \left(p ∧ q\right) is false when pp is true and qq is false

Hence, not a tautology
Also, p⇒(p⇔q)p ⇒ \left(p ⇔ q\right) is false when pp is true and qq is false

Hence, not a tautology

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