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

Answer: (A)

$(pΔr)∨q$

Answer 2

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).