## Answers to Selected Problems on Combinational Logic

(See Chapter 4 of Mano's Digital Design (2nd ed.))

 4-4 Design a combinational circuit that adds one to a 4-bit binary number. For example, if the input of the circuit is 1101, the output is 1110. The circuit can be designed using four half-adders. Let A3A2A1A0 + 1 = S4S3S2S1S0 4-5 A combinational circuit produces the binary sum of two 2-bit numbers, x1x0 and y1y0. The outputs are C, S1, and S0. Provide a truth tableof the combinational circuit. 4-6 Design a circuit for the above problem using two full-adders. 4-18 Derive the truth table of the circuit shown below. F1 = A'B'C + A'BC' + AB'C' + ABC F2 = AB + BC + AC (Note: it is a full adder). 4-19 Draw the NAND logic diagram for each of the following expressions using multiple-level NAND gate circuits: (a) (AB' + CD')E + BC(A + B) = (AB' + CD')E + BC(A'B')'   (b) w(x + y + z) + xyz = w( x'y'z') + xyz 4-20 Convert the logic diagram of the code converter shown in Fig. 4-8 to a multiple-level NAND circuit. 4-22 Verify that the circuit shown below generates the exclusive-NOR function. 4-24 Derive the truth table for the output of each NOR gate shown below. 4-25 Prove the following equality. x'y = (x')'y + x'y' = xy +x'y' = (x'y + xy')' = ( xy ) = x'y' + xy = x'(y') + x(y')' = xy' 4-26 Prove the following equality. x1 = x'•1 + x•1' = x'•1 + x•0= x'+ 0 = x' x0 = x'•0 + x•0' = x'•0 + x•1= 0 + x = x 4-27 Show that if xy = 0 then x XOR y is equal to x+y. If xy = 0 then xy = (x'y' + xy)' = (x'y' + 0)' = x + y 4-29 Design the circuit of a 3-bit parity generator and the circuit of a 4-bit parity generator using an odd parity bit.

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