Using the equivalence property, show that p ↔️ q ≡ ( p ∧ q) v (ㄱp ∧ ㄱq)

Verify whether the following compound propositions are tautologies or contradictions or contingency
((p⟶ q) ∧ (q ⟶ r)) ⟶ (p ⟶ r)

Let M = \(\left\{ \left( \begin{matrix} x & x \\ x & x \end{matrix} \right) :x\in R-\{ 0\} \right\} \) and let * be the matrix multiplication. Determine whether M is closed under ∗. If so, examine the commutative and associative properties satisfied by ∗ on M.

Let A be Q\{1}. Define ∗ on A by x*y = x + y − xy. Is ∗ binary on A? If so, examine the commutative and associative properties satisfied by ∗ on A.

Check whether the statement p➝(q➝p) is a tautology or a contradiction without using the truth table.