Simplify the following Boolean function by a four-variable K-map in terms of minimum Sum of Products and minimum Product of Sums a. F(A,B,C,D)= Σm(0, 2, 5, 8, 9, 11, 12, 13) b. F(A,B,C,D)=Σm(0, 2,3, 5, 7, 8, 10, 11, 14, 15)

Using K-maps, simplify the following Boolean functions and express each of them in sum-of- products (SOP) and product-of-sums (POS) forms. a) F(w, x, y, z) = Σ(0, 1, 2, 5, 8, 10, 13) Σ(1, 3, 6, 7, 8, 9, 12, 13, 14, 15) Σ(2, 4, 10, 12, 14) and d (w, x, y, z) = (0,1,5,8) (0, 4, 8, 9, 10, 11, 12, 14) b) F(w, x, y, z) = c) F(w, x, y, z) = d) F(w, x, y, z) =

Using Karnaugh Maps, simplify the following Boolean Expression: T = ABC + AC + AB + ABC +ĀCD + ABD given that B cannot be 1 if C is 0.

Prove that if 3x(P(x) → Q(x) is true, then VxP(x) → 3xQ(x) is true.