Prove by using derivation rules that (A' rightarrow B') rightarrow (B rightarrow A) is a tautology. Prove by using derivation rules that (A logicaland B)' logicaland A rightarrow B' is a tautology. If P(x) = x/2 is even, for (there exists x)P(x) logicaland (for all x) [P(x)]' on DOI = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20} list the values for which it is valid. If D(x, y) = x divides y, for (for all x)(there exist y)D(x, y) on DOI = D_x times D_y = {{2, 3} times {4, 6}} list the values for which it is valid. Translate the English statements into a predicate wff. Everyone laughs, but no one knows why. Use L(x) and K(x)