Abstract Algebra Theory and Applications - Computer Science ...

EXERCISES 309aa b ca c ′ aa ′ bbc ′9. Let X be a finite set containing n elements. Prove that P(X) = 2 n . Concludethat the order of any finite Boolean algebra must be 2 n for some n ∈ N.10. For each of the following circuits, write a Boolean expression. If the circuitcan be replaced by one with fewer switches, give the Boolean expression anddraw a diagram for the new circuit.a ′ a b ′baaba ′ba ′ba b ca ′ b ′ ca b ′ c ′11. Prove or disprove: The set of all nonzero integers is a lattice, where a ≼ b isdefined by a | b.12. Prove that a ∧ b is the greatest lower bound of a and b in Theorem 17.3.13. Let L be a nonempty set with two binary operations ∨ and ∧ satisfying thecommutative, associative, idempotent, and absorption laws. We can define apartial order on L, as in Theorem 17.3, by a ≼ b if a ∨ b = b. Prove that thegreatest lower bound of a and b is a ∧ b.14. Let G be a group and X be the set of subgroups of G ordered by set-theoreticinclusion. If H and K are subgroups of G, show that the least upper boundof H and K is the subgroup generated by H ∪ K.