Answers to Selected Problems in Chapter 4, COSC3410

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.

COSC 3410 Answers to Selected Problems, Chapter | 1 | 2 | 3 | 4 | 5 | 6 | 7 |

visits since 4/13/98. RETURN to J. C. Huang's home page.

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.