A computer interface to a line printer has seven data

Represent each of the following sentences by a Boolean equation. (a) The company safe should be unlocked only when Mr. Jones is in the office or Mr. Evans is in the office, and only when the company is open for business, and only when the security guard is present. (b) You should wear your overshoes if you are outside in a heavy rain and you are wearing your new suede shoes, or if your mother tells you to. (c) You should laugh at a joke if it is funny, it is in good taste, and it is not offensive to others, or if it is told in class by your professor (regardless of whether it is funny and in good taste) and it is not offensive to others. (d) The elevator door should open if the elevator is stopped, it is level with the floor, and the timer has not expired, or if the elevator is stopped, it is level with the floor, and a button is pressed.

A flow rate sensing device used on a liquid transport pipeline functions as follows. The device provides a 5-bit output where all five bits are zero if the flow rate is less than 10 gallons per minute. The first bit is 1 if the flow rate is at least 10 gallons per minute; the first and second bits are 1 if the flow rate is at least 20 gallons per minute; the first, second, and third bits are 1 if the flow rate is at least 30 gallons per minute; and so on. The five bits, represented by the logical variables A, B, C, D, and E, are used as inputs to a device that provides two outputs Y and Z. (a) Write an equation for the output Y if we want Y to be 1 iff the flow rate is less than 30 gallons per minute. (b) Write an equation for the output Z if we want Z to be 1 iff the flow rate is at least 20 gallons per minute but less than 50 gallons per minute.

Given \(F_{1}=\Sigma m(0,4,5,6)\) and \(F_{2}=\Sigma m(0,3,6,7)\) find the minterm expression for \(F_1 + F_2\). State a general rule for finding the expression for \(F_1 + F_2\) given the minterm expansions for \(F_1\) and \(F_2\). Prove your answer by using the general form of the minterm expansion.

(a) How many switching functions of two variables (x and y) are there? (b) Give each function in truth table form and in reduced algebraic form.

A combinational circuit is divided into two subcircuits \(N_1\) and \(N_2\) as shown. The truth table for \(N_1\) is given. Assume that the input combinations ABC = 110 and ABC = 010 never occur. Change as many of the values of D, E, and F to don’t-cares as you can without changing the value of the output Z.

Work (a) and (b) with the following truth table: (a) Find the simplest expression for F, and specify the values of the don’t-cares that lead to this expression. (b) Repeat (a) for G. (Hint: Can you make G the same as one of the inputs by properly choosing the values for the don’t-care?)

Each of three coins has two sides, heads and tails. Represent the heads or tails status of each coin by a logical variable (A for the first coin, B for the second coin, and C for the third) where the logical variable is 1 for heads and 0 for tails. Write a logic function F(A, B, C) which is 1 iff exactly one of the coins is heads after a toss of the coins. Express F (a) as a minterm expansion. (b) as a maxterm expansion.

A switching circuit has four inputs as shown. A and B represent the first and second bits of a binary number \(N_1\). C and D represent the first and second bits of a binary number \(N_2\). The output is to be 1 only if the product \(N_1 \times N_2\) is less than or equal to 2. (a) Find the minterm expansion for F. (b) Find the maxterm expansion for F. Express your answers in both decimal notation and algebraic form.

Given: \(F(a, b, c) = abc^\prime + b^\prime\). (a) Express F as a minterm expansion. (Use m-notation.) (b) Express F as a maxterm expansion. (Use M-notation.) (c) Express \(F^\prime\) as a minterm expansion. (Use m-notation.) (d) Express \(F^\prime\) as a maxterm expansion. (Use M-notation.)

Work Problem 4.9 using: \(F(a, b, c, d)=(a+b+d)\left(a^{\prime}+c\right)\left(a^{\prime}+b^{\prime}+c^{\prime}\right)\left(a+b+c^{\prime}+d^{\prime}\right)\)

(a) Implement a full subtracter using a minimum number of gates. (b) Compare the logic equations for the full adder and full subtracter. What is the relation between \(s_i\) and \(d_i\)? Between \(c_{i+1}\) and \(b_{i+1}\)?

Design a circuit which will perform the following function on three 4-bit numbers: \(\left(X_{3} X_{2} X_{1} X_{0}+Y_{3} Y_{2} Y_{1} Y_{0}\right)-Z_{3} Z_{2} Z_{1} Z_{0}\) It will give a result \(S_3 S_2 S_1 S_0\), a carry, and a borrow. Use eight full adders and any other type of gates. Assume that negative numbers are represented in 2’s complement.

A combinational logic circuit has four inputs (A, B, C, and D) and one output Z. The output is 1 iff the input has three consecutive 0’s or three consecutive 1’s. For example, if A = 1, B = 0, C = 0, and D = 0, then Z = 1, but if A = 0, B = 1, C = 0, and D = 0, then Z = 0. Design the circuit using one four-input OR gate and four three-input AND gates.

Design a combinational logic circuit which has one output Z and a 4-bit input ABCD representing a binary number. Z should be 1 iff the input is at least 5, but is no greater than 11. Use one OR gate (three inputs) and three AND gates (with no more than three inputs each).

A logic circuit realizing the function f has four inputs A, B, C, and D. The three inputs A, B, and C are the binary representation of the digits 0 through 7 with A being the most-significant bit. The input D is an odd-parity bit, i.e., the value of D is such that A, B, C, and D always contain an odd number of 1’s. (For example, the digit 1 is represented by ABC = 001 and D = 0, and the digit 3 is represented by ABCD = 0111.) The function f has value 1 if the input digit is a prime number. (A number is prime if it is divisible only by itself and 1; 1 is considered to be prime and 0 is not.) (a) List the minterms and don’t-care minterms of f in algebraic form. (b) List the maxterms and don’t-care maxterms of f in algebraic form.

A priority encoder circuit has four inputs, \(x_3\), \(x_2\), \(x_1\), and \(x_0\). The circuit has three outputs: z, \(y_{1}\), and \(y_{0}\). If one of the inputs is 1, z is 1 and \(y_1\) and \(y_0\) represent a 2-bit, binary number whose value equals the index of the highest numbered input that is 1. For example, if \(x_2\) is 1 and \(x_3\) is 0, then the outputs are z = 1 and \(y_1 = 1\) and \(y_0 = 0\). If all inputs are 0, z = 0 and \(y_1\) and \(y_0\) are don’t-cares. (a) List in decimal form the minterms and don’t-care minterms of each output. (b) List in decimal form the maxterms and don’t-care maxterms of each output.

The 9’s complement of a decimal digit d (0 to 9) is defined to be 9 ? d. A logic circuit produces the 9’s complement of an input digit where the input and output digits are represented in BCD. Label the inputs A, B, C, and D, and label the outputs W, X, Y and Z. (a) Determine the minterms and don’t-care minterms for each of the outputs. (b) Determine the maxterms and don’t-care maxterms for each of the outputs.

Repeat Problem 4.17 for the case where the input and output digits are represented using the 4-2-2-1 weighted code. (If only one weight of 2 is required for decimal digits less than 5, select the rightmost 2. In addition, select the codes so that \(W = A^\prime\), \(X = B^\prime\), \(Y = C^\prime\), and \(Z = D^\prime\). (There are two possible codes with these restrictions.)

Each of the following sentences has two possible interpretations depending on whether the AND or OR is done first. Write an equation for each interpretation. (a) The buzzer will sound if the key is in the ignition switch, and the car door is open, or the seat belts are not fastened. (b) You will gain weight if you eat too much, or you do not exercise enough, and your metabolism rate is too low. (c) The speaker will be damaged if the volume is set too high, and loud music is played, or the stereo is too powerful. (d) The roads will be very slippery if it snows, or it rains, and there is oil on the road.

A bank vault has three locks with a different key for each lock. Each key is owned by a different person. To open the door, at least two people must insert their keys into the assigned locks. The signal lines A, B, and C are 1 if there is a key inserted into lock 1, 2, or 3, respectively. Write an equation for the variable Z which is 1 iff the door should open.

A paper tape reader used as an input device to a computer has five rows of holes as shown. A hole punched in the tape indicates a logic 1, and no hole indicates a logic 0. As each hole pattern passes under the photocells, the pattern is translated into logic signals on lines A, B, C, D, and E. All patterns of holes indicate a valid character with two exceptions. A pattern consisting of none of the possible holes punched is not used because it is impossible to distinguish between this pattern and the unpunched space between patterns. An incorrect pattern punched on the tape is erased by punching all five holes in that position. Therefore, a valid character punched on the tape will have at least one hole but will not have all five holes punched. (a) Write an equation for a variable Z which is 1 iff a valid character is being read. (b) Write an equation for a variable Y which is 1 iff the hole pattern being read has holes punched only in rows C and E.

A computer interface to a line printer has seven data lines that control the movement of the paper and the print head and determine which character to print. The data lines are labeled A, B, C, D, E, F, and G, and each represents a binary 0 or 1. When the data lines are interpreted as a 7-bit binary number with line A being the most significant bit, the data lines can represent the numbers 0 to \(127_{10}\). The number \(13_{10}\) is the command to return the print head to the beginning of a line, the number \(10_{10}\) means to advance the paper by one line, and the numbers \(32_{10}\) to \(127_{10}\) represent printing characters. (a) Write an equation for the variable X which is 1 iff the data lines indicate a command to return the print head to the beginning of the line. (b) Write an equation for the variable Y which is 1 iff there is an advance paper command on the data lines. (c) Write an equation for the variable Z which is 1 iff the data lines indicate a printable character. (Hint: Consider the binary representations of the numbers 0–31 and 32–127 and write the equation for Z with only two terms.)

Given \(F_{1}=\Pi M(0,4,5,6)\) and \(F_2 = \Pi M(0, 4, 7)\), find the maxterm expansion for \(F_1 F_2\). State a general rule for finding the maxterm expansion of \(F_{1} F_{2}\) given the maxterm expansions of \(F_1\) and \(F_2\). Prove your answer by using the general form of the maxterm expansion.

Given \(F_1 = \Pi M(0, 4, 5, 6)\) and \(F_2 = \Pi M(0, 4, 7)\), find the maxterm expansion for \(F_1 + F_2\). State a general rule for finding the maxterm expansion of \(F_1 + F_2\), given the maxterm expansions of \(F_1\) and \(F_2\). Prove your answer by using the general form of the maxterm expansion.

Four chairs are placed in a row: Each chair may be occupied (1) or empty (0). Give the minterm and maxterm expansion for each logic function described. (a) F(A, B, C, D) is 1 iff there are no adjacent empty chairs. (b) G(A, B, C, D) is 1 iff the chairs on the ends are both empty. (c) H(A, B, C, D) is 1 iff at least three chairs are full. (d) J(A, B, C, D) is 1 iff there are more people sitting in the left two chairs than in the right two chairs.

Four chairs (A, B, C, and D) are placed in a circle: A next to B, B next to C, C next to D, and D next to A. Each chair may be occupied (1) or empty (0). Give the minterm and maxterm expansion for each of the following logic functions: (a) F(A, B, C, D) is 1 iff there are no adjacent empty chairs. (b) G(A, B, C, D) is 1 iff there are at least three adjacent empty chairs. (c) H(A, B, C, D) is 1 iff at least three chairs are full. (d) J(A, B, C, D) is 1 iff there are more people sitting in chairs A and B than chairs C and D.

Given \(f(a, b, c) = a(b + c^\prime)\). (a) Express f as a minterm expansion (use m-notation). (b) Express f as maxterm expansion (use M-notation). (c) Express \(f ^\prime\) as a minterm expansion (use m-notation). (d) Express \(f^\prime\) as a maxterm expansion (use M-notation).

Work Problem 4.27 using \(f(a, b, c, d)=a c d+b d^{\prime}+a^{\prime} c^{\prime} d+a b^{\prime} c d+a^{\prime} b^{\prime} c d^{\prime}\).

Find both the minterm expansion and maxterm expansion for the following functions, using algebraic manipulations: (a) \(f(A, B, C, D)=A B+A^{\prime} C D\) (b) \(f(A, B, C, D) = (A + B + D^\prime)(A^\prime + C)(C + D) \)

Given \(F^{\prime}(A, B, C, D)=\Sigma m(0,1,2,6,7,13,15)\). (a) Find the minterm expansion for F (both decimal and algebraic form). (b) Find the maxterm expansion for F (both decimal and algebraic form).

Repeat Problem 4.30 for \(F^{\prime}(A, B, C, D)=\Sigma m(1,2,5,6,10,15)\).

Work parts (a) through (d) with the given truth table. (a) Find the simplest expression for \(F_1\), and specify the values for the don’t-cares that lead to this expression. (b) Repeat for \(F_2\). (c) Repeat for \(F_3\). (d) Repeat for \(F_4\).

Work Problem 4.5 using the following circuits and truth table. Assume that the input combinations of ABC = 011 and ABC = 110 will never occur.

Work Problem 4.7 for the following logic functions: (a) \(G_1(A, B, C)\) is 1 iff all the coins landed on the same side (heads or tails). (b) \(G_2(A, B, C)\) is 1 iff the second coin landed on the same side as the first coin.

A combinational circuit has four inputs (A, B, C, D) and three outputs (X, Y, Z). XYZ represents a binary number whose value equals the number of 1’s at the input. For example if ABCD = 1011, XYZ = 011. (a) Find the minterm expansions for X, Y, and Z. (b) Find the maxterm expansions for Y and Z.

A combinational circuit has four inputs (A, B, C, D) and four outputs (W, X, Y, Z). WXYZ represents an excess-3 coded number whose value equals the number of 1’s at the input. For example, if ABCD = 1101, WXYZ = 0110. (a) Find the minterm expansions for X, Y, and Z. (b) Find the maxterm expansions for Y and Z.

A combinational circuit has four inputs (A, B, C, D), which represent a binary-coded-decimal digit. The circuit has two groups of four outputs—S, T, U, V, and W, X, Y, Z. Each group represents a BCD digit. The output digits represent a decimal number which is five times the input number. For example, if ABCD = 0111, the outputs are 0011 0101. Assume that invalid BCD digits do not occur as inputs. (a) Construct the truth table. (b) Write down the minimum expressions for the outputs by inspection of the truth table. (Hint: Try to match output columns in the table with input columns.)

Work Problem 4.37 where the BCD outputs represent a decimal number that is 1 more than four times the input number. For example, if ABCD = 0011, the outputs are 0001 0011.

Design a circuit which will add a 4-bit binary number to a 5-bit binary number. Use five full adders. Assume negative numbers are represented in 2’s complement. (Hint: How do you make a 4-bit binary number into a 5-bit binary number, without making a negative number positive or a positive number negative? Try writing down the representation for –3 as a 3-bit 2’s complement number, a 4-bit 2’s complement number, and a 5-bit 2’s complement number. Recall that one way to find the 2’s complement of a binary number is to complement all bits to the left of the first 1.)

A half adder is a circuit that adds two bits to give a sum and a carry. Give the truth table for a half adder, and design the circuit using only two gates. Then design a circuit which will find the 2’s complement of a 4-bit binary number. Use four half adders and any additional gates. (Hint: Recall that one way to find the 2’s complement of a binary number is to complement all bits, and then add 1.)

(a) Write the switching function f(x, y) = x + y as a sum of minterms and as a product of maxterms. (b) Consider the Boolean algebra of four elements {0, 1, a, b} specified by the following operation tables and the Boolean function f(x, y) = ax + by where a and b are two of the elements in the Boolean algebra. Write f(x, y) in a sum-of-minterms form. (c) Write the Boolean function of part (b) in a product-of-maxterms form. (d) Give a table of combinations for the Boolean function of part (b). (Note: The table of combinations has 16 rows, not just 4.) (e) Which four rows of the table of combinations completely specify the function of part (b)? Verify your answer.

(a) If \(m_1\) and \(m_2\) are minterms of n variables, prove that \(m_1 + m_2 = m_1 \oplus m_2\). (b) Prove that any switching function can be written as the exclusive OR sum of products where each product does not contain a complemented literal. (Hint: Start with the function written as a sum of minterms and use part (a).)

(a) Show that the full adder of Figure 4-5 can be implemented using two 2-input exclusive OR gates and three 2-input NAND gates. (Hint: Rewrite Equation (4-2) in terms of \(X \oplus Y\).) (b) Compare the maximum addition time of the ripple-carry adder of Figure 4-3 using the full adder of part (a) versus the full adder of Figure 4-5 assuming the same gate types are used in both.

Show that a full subtractor can be implemented using two 2-input exclusive OR gates, one inverter, and three 2-input NOR gates. (Hint: Write the borrow out equation in product-of-sums form.)

The full adder of Figure 4-5 is modified by adding two control inputs, \(E_1\) and \(E_0\), and implemented as shown in the figure below. (a) For each combination of values for \(E_1\) and \(E_0\), give the algebraic expression for the outputs of the full adder. (b) Assume this modified full adder is used in the parallel adder of Figure 4-3. For each combination of values for \(E_{1}\) and\(E_{0}\), specify the function (Add, Exclusive OR, etc.) performed by the parallel adder.

Redo Problem 4.45 if gates \(G_1\) and \(G_2\) are NAND gates rather than AND gates.

Redo Problem 4.45 if gates \(G_1\) and \(G_2\) are NOR gates rather than AND gates and an inverter is inserted in the \(c_i\) input of \(G_2\).

Redo Problem 4.45 if gates \(G_1\) and \(G_2\) are OR gates rather than AND gates and an inverter is inserted in the \(c_i\) input of \(G_2\).