please show as much work as possible

PROBLEM 1 (28 PTS) a) Simplify the following functions using ONLY Boolean Algebra Theorems. For each resulting simplified function, sketch the logic circuit using AND, OR, XOR, and NOT gates. (15 pts) F(x, y, z) = 11(M.,M ,M4, M) F = x(yz) + y F = (AC + D) (A + C + D) b) Determine whether or not the following expression is valid, i.e., whether the left and right-hand sides represent the same function. Suggestion: complete the truth tables for both sides: (5 pts) X1X2 + X,X3 + X2 X3 = X1 X3 + x2x3 + x, X2 , 2 C) For the following Truth table with two outputs: (8 pts) 0 0 0 Provide the Boolean functions using the Canonical Sum of Products (SOP), and Product of Sums (POS). (4 pts) Express the Boolean functions using the minterms and maxterms representations. Sketch the logic circuits as Canonical Sum of Products and Product of Sums. (3 pts) 0 0 0 0 1 0 1 1 1 1 F