1.Determine if the following argument is valid.Use truth tables.

If this car is made in England, then parts are hard to get.This car is expensive, or it is not difficult to obtain parts.But this car is not expensive.Hence, it is not made in England.

M - Made in England

P - Parts are hard to get

E - Car Expensive

2. A and B are nxn matrices such that AB = BA = 0.Prove that (A+B)^k = A^k + B^k for all k's in Z⁺ .Use any method of proof.

3. For all sets A and B, Prove or disprove: P(AxB) = P(A) x P(B).(P (X) is the power set of X).

4. For some set S, let A, B be elements of P(S).Define A R B if the symmetric difference of sets A and B is finite.Prove that R is an equivalence relation.