[2.1] Given the pulse train sent to the input A of an OR gate, draw its output pulse train. 0910 :-D [2.2] Given the pulse train sent to the input A, B and C of an OR gate, draw its output pulse train. A1 1 1 1 1 . 1 1 1 1 1 1 0 1 2.3] The output of an AND gate shows the following pulse train. Draw the corresponding signal at input A. 11110 4D [2.4 Given the following pulse trains sent to the 2 inputs A and B of an AND gate, find the output pulse train. 1 0 1 0 [2.5] Given the following pulse trains sent to the 3 inputs A, B and C of an AND gate, find the output pulse train. 0 . 1 B D 1 o 1 0 1 C 1 + 1 o 1 1 o 1 [2.6] The output Y and the inputs A and C of an AND gate have the following pulse trains. Draw the corresponding signal at input B. 1 o D. 0 1 1 0 1 + 0 1 1 1 o 1 12.7) Given the following pulse trains sent to the 3 inputs A, B and C of a NOR gate, find the output pulse train. 0 O 0 1 1 0 o 0 R 1 1 . . + 0 1 1 1 . 1 1 0 1 [2.8] Given the following pulse trains sent to the 3 inputs A, B and C of a NAND gate, find the output pulse train. B1 1 . C1 1 1 1 . + [2.9] Demonstrate the validity of the following identities by means of truth tables: (a) De Morgan's theorem for three variables: (x + y + 2) = xyz and (xyz) = 8 ++z (b) The distributive law: x + y2 = x +yX x+z) (c) The distributive law: x(y + 2) = xy + x2 (d) The associative law: x + (y +2) = (x+y)+z (e) The associative law: x(yz) - (xyz [2.10] Simplify the following Boolean expressions to a minimum number of literals: (a) xy + xy (b) (x + y)(x + y) (c) xyz + xy + xyz (d) (A + B) (A + B) (e) (a + b + c)(ab+c) [2.11) Simplify the following Boolean expressions to a minimum number of literals: (a) ABC + AB + ABC (b) fyz + xz (c) (BC + D) (AB + CD) (d) xy + x(wz + w2) [2.12] Reduce the following Boolean expressions to the indicated number of literals: (a) AC + ABC + AC to three literals. (b) (x9 + 2) + 2 + xy +wx to three literals. @ B(D+CD) + B(A +CD) to one literals. d) (+C)(+C)(A+B+CD) to four literals. [2.13] Draw logic diagrams of the circuits that implement the original and simplified expressions in Problem 2.11. [2.14] Find the complement of the following expressions : (a) xy + xy (b)(a + c)(a + b)(a + b + c) (c) z+zlw + xy) 2.15] We can perform logical operations on strings of bits by considering each pair of corresponding bits separately called bitwise operation). Given two eight-bit strings A =10110001 and B =10101100, evaluate the eight-bit result after the following logical operations : (a) AND (b) OR (C) XOR [2.16] Draw logic diagrams to implement the following Boolean expressions : (a) f = [(u + 7) +z)] (b) f = (u y) + x (c) f = ( + x)(y + 2) a f = u + x + x(u+y) [2.17] Implement the Boolean function F = xy + xy + yz (a) With AND, OR, and inverter gates (b) With OR and inverter gates (c) With AND and inverter gates (d) With NAND and inverter gates (e) With NOR and inverter gates [2.18] Simplify the following Boolean functions T, and T, to a minimum number of literals: 0 0 0 0 1 1 1 1 B 0 0 1 1 0 0 1 1 C 0 1 0 1 0 1 0 1 T 1 1 1 0 0 0 0 0 T, 0 0 0 1 1 1 1 1 [2.19) Obtain the truth table of the following functions, and express each function in sum-of-minterms and product-of-maxterms form: (a)(b + cd)(c + bd) (b) (cd + bc + b)(b + d) [2.20] For the Boolean function F = xyz + xyz + wiy + wxy + wxy (a) Obtain the truth table of F. (b) Draw the logic diagram, using the original Boolean expression (c) Use Boolean algebra to simplify the function to a minimum number of literals. [2.21] Express the following function as a sum of minterms and as a product of maxterms: F(A,B,C,D) = BD +D+ BD [2.22] Express the complement of the following functions in sum-of-minterms form: (a) F(A,B,C,D) = (2, 4, 7, 10, 12, 14) (b)F(x,y,z) = [(3, 5, 7) [2.23] Convert each of the following to the other canonical form: (a) F(x,y,z) = {(1,3,5) (b)F(A,B,C,D) = |(3, 5, 8, 11) [2.24] Convert each of the following expressions into sum of products and product of sums : (a) (u + xw)(x + v) (b) x + x(x + y)(y + 2) [2.25] Draw the logic diagram corresponding to the following Boolean expressions without simplifying them: (a) BC + AB + ACD (b) (A + B)(C+ D) (A + B + D) (AB + AB)(CD + CD) (d) A + CD + (A +D)(C+D) [2.26) Write the following Boolean expressions in sum of products form: (b + d)(a + b + c) [2.27] Write the following Boolean expression in product of sums form: b+c + abc