Digital Electronics Tutorial Sheet 2BOOLEAN ALGEBRA & K-MAP

These questions are intended to:• involve you in active study• help you understand and practice techniques• indicate what you are expected to be able to do / understand• give you feedback on how much you understand

Qus.1: Find the complement of the functionf= C(AB + A’ B’D + A’BD’), i) Using DeMorgan’s Theorem ii) By taking the dual and complementing each literal

Qus.2: Prove the following identities algebraically and by means of truth tables: (a) (A + B ) ( A+B)’ = 0 (b) XY + X’Y’ +XY’+X’Y=1 (c) (A+A’B)’=A’B’ (d) (X’+Y)(X+Y’)=(X Y)’ Qus.3: Using DeMorgan’s theorem, draw logic diagrams for F = ABC’+A’B’+BC (a) Using only AND gates and inverters. (b) Using only OR gates and inverters You may use two-input and three-input AND and OR gates for (a) and (b).Qus.4: Express each of the following expressions in terms of minterms and maxterms. (a)F=BC’+A’B+B(A+C) (b)F=(A+B’+C)(A’+B)Qus.5: Simplify each of the following Boolean expressions as much as possible using identities: (a)ABC’+AB(CD)’+ABD’ (b)BC+ABCD’+A’BCD+ABCD (c)(X’+Y’)(XY)’+XYZ+XYZ’Qus.6: Simplify each of the following functions for F using a K-map. (a)F(W, X, Y, Z) =∑ m(0, 1,4, 5, 8, 9) (b)F(A, B, C, D) =∑m(O, 2, 8, 10, 12, 14) (c) F(A, B, C, D) = ∑ m(2,4, 5,6, 7, 10, 14)

(d)F(W,X, Y, Z) =∑rn(2,3,6,7, 8,9, 12, 13)Qus.7: Minimize each of the following expressions for F’ using a K-map in sums-of-productform:

(a) F(W,X, Y,Z) =W’X’YZ+WYZ(b) F= A’B’C’D’+A’CD+ABCD(c) (A’+B’+C+D’)(A’+B+C+D’)(A+B’+C+D’)

Qus.8: Find essential prime implicants and then minimize each of the following functions for F using a K-map:

(a) F(W,X,Y,Z)=∑m(2,3,6,7,8,9,12,13,15) (b)F(A, B, C, D) = ∑ m(3,4, 5,7, 11, 12,15)Qus.9: Minimize each of the following functions for f using a K-map and don't care Conditions d. (a) f ( A , B, C, D) = ∑m(0,2,3, 11) d(A, B, C, D) =∑m(1, 8,9, 10) (b) f(A,B,C,D)=∑m(4,5,10,11) d(A, B, C, D) =∑m(1 2 , 13, 14, 15)Qus.10: Minimize the following expression using a K-map: F=AB+A’B’C’D’+CD’A’B’C’D and then draw schematics using: (a)NAND gates. (b) NOR gates.Qus.11: Four chairs A, B, C, and D are placed in a row. Each chair may be occupied (“1“) or empty (“0“). A Boolean function F is (“1“) if and only if there are two or more adjacent chairs that are empty. a) Give the truth table defining the Boolean F. b) Express F as a minterm expansion (standard sum of products). c) Express F as a maxterm expansion (standard product of sums). d) Simplify the minterm expansion of FQus.12: A buzzer is to sound when the following conditions apply: ●Switches A, B, C are on. ●Switches A and B are on but switch C is off. ● Switches A and C are on but switch B is off. ● Switches B and C are on but switch A is off. a) Draw a truth table for this situation and obtain a Boolean expression for it. b) Minimize this expression and draw a logic diagram for it