Question

Let Δ, ∇ ∈ {∧, ∨} be such that p ∇ q ⇒ ((p ∇ q) ∇ r) is a tautology. Then (p ∇ q) Δ r is logically equivalent to ______.

Options

• (p Δ r) ∨ q

• (p Δ r) ∧ q

• (p ∧ r) Δ q

• (p ∇ r) ∧ q

Answer 1

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

Explanation:

Case-I : If Δ ≡ ∇ ≡ ∨

(p ∨ q) `rightarrow` ((p ∨ q) ∨ r) ≡ tautology

Then (p ∨ q) ∨ r ≡ (p Δ r) ∨ q

Case-II : If Δ ≡ ∇ ≡ ∧

(p ∧ q) `rightarrow` ((p ∧ q) ∧ r)

It will be false if r is false.

So not a tautology

Case-III : If Δ ≡ ∨, ∇ ≡ ∧

Then (p ∧ q) `rightarrow` {(p ∨ q) ∧ r}

Not a tautology

(Check p `rightarrow` T, q `rightarrow` T, r `rightarrow` F)

Case-IV : If Δ ≡ ∧, ∇ ≡ ∨

(p ∧ q) `rightarrow` {(p ∧ q) ∨ r}

Not a tautology