Question

Let\( Δ,▽∈{∧,∨} \)
be such that \(p▽q⇒((pΔq)▽r) \)
is a tautology. Then \((p▽q)Δr \)
is logically equivalent to:

• A. \((pΔr)∨q\)
• B. \((pΔr)∧q\)
• C. \((p∧r)Δq\)
• D. \((p▽r)∧q\)

Answer 1

The Correct Option is A

The correct answer is (A) : \((pΔr)∨q\)
Case-I If ∇ is same as ∧
Then (p∧q) ⇒ ((pΔq) ∧r) is equivalent to ~ (p∧q) ∨ ((pΔq) ∧r) is equivalent to (~ (p∧q) ∨ (pΔq))∧ (~ (p∧q) ∨r)
Which cannot be a tautology
For both Δ (i.e.∨ or ∧)

Case-II If ∇ is same as ∨
Then (p∨q) ⇒ ((pΔq) ∨r) is equivalent to
~(p∨q) ∨ (pΔq) ∨r which can be a tautology if Δ is also same as ∨.
Hence both Δ and ∇ are same as ∨.
Now (p∇q) Δr is equivalent to (p∨q∨r).