1. Find the minimum cost implementation in SOP (Sum of Products) and POS (Product of Sums)for the function f(x1,x2,x3) = Σ m(1,2,3,5). Using the formula for determining the cost of an implementation, calculate the cost.

1. Find the minimum cost implementation in SOP (Sum of Products) and POS (Product of Sums)for the function f(x1,x2,x3) = Σ m(1,2,3,5). Using the formula for determining the cost of an implementation, calculate the cost.
2. A four variable logic function that is equal to 1 if any three or all four of its variables are equal to 1 is called majority function. Design a minimum cost SOP circuit that implements this majority function.
3. Find the minimal SOP expression for f(x1,x2,x3, x4) = Σ m(1,5,7,9,11,15) 4. Find the minimum sum-of-product of the following functions:
a. F=X’Z + XY + XY’Z b. F=A’C’D + B’CD + AC’D+BCD c. F=WXZ’ + WX’YZ + XZ d. F=ABC’D’ +A’BC’+ABD+A’CD+BCD’
5. A circuit with two outputs has to implement the following functions: f(x1,..,x4) = Σ m(0,2,4,6,7,9) + D(10,11) g(x1,..,x4)= Σ m(2,4,9,10,15)+D(0,13,14)
Design the minimum-cost circuit and compare its cost with combined costs of two circuits that implements f and g separately. Assume that the input variables are available in both complemented and uncomplemented forms.
NOTE: Calculate the cost of a logic circuit as the number of gates plus the total number of inputs to all gates in the circuit. Assuming that inputs are available both true and complemented form at zero cost. If an inversion is needed inside a circuit, then the corresponding NOT gate and its input are included in the cost.