Let Δ, ∇ ∈ \ ∧ , ∨\ be such that p ∇ q → ((p ) ∇ r) is a tautology....

Case-I

If $\Delta ≡ ∇ ≡ ∧$

\left(p ∧ q\right) \to \left(\left(p ∧ q\right) ∧ r\right)$</p><p>then it is false if$r$is false.</p><p>So not a tautology</p><p>Case-II</p><p>If$\Delta ≡ ∇ ≡ ∨$</p><p>\left(p ∨ q\right) \to \left(\left(p ∨ q\right) ∨ r\right) ≡$ tautology

then \left(p ∨ q\right) ∨ r ≡ \left(p\Deltar\right) ∨ q$</p><p>Case-III</p><p>If$\Delta = ∨ , ∇ = ∧$</p><p>then \left(p ∧ q\right) \to \left\{\left(p ∨ q\right) ∧ r\right\}$ is not a tautology

when $p \to T, q \to T, r \to F$

Case-IV

If $\Delta = ∧ , ∇ = ∨$

\left(p ∧ q\right) \to \left\{\left(p ∧ q\right) ∨ r\right\} is not a tautology