6.1 Combinatorial Circuits

Basic Skills

1. A circuit has two inputs, [latex]A[/latex] and [latex]B[/latex], and produces an output of 1 only when both [latex]A[/latex] and [latex]B[/latex] are [latex]1[/latex]. What type of logic gate is this?
2. Construct the truth table for a circuit with two inputs, [latex]A[/latex] and [latex]B[/latex], and an output defined by the expression [latex]A + B[/latex].
3. Given a circuit with inputs [latex]A = 0[/latex] and [latex]B = 1[/latex], and the output is defined by [latex]A \cdot B[/latex], what is the output?
4. What is the output of a NOT gate if the input is 0? What is the output if the input is 1?
5. A circuit has three inputs: [latex]A[/latex], [latex]B[/latex], and [latex]C[/latex]. The output is defined by the expression [latex](A \cdot B) + C[/latex]. Evaluate the output when [latex]A = 1[/latex], [latex]B = 0[/latex], and [latex]C = 1[/latex].

Applications

6. A smart home lighting system turns on the lights if either the motion sensor detects movement or the light sensor detects darkness. Let [latex]M[/latex]: “Motion is detected” and [latex]D[/latex]: “Darkness is detected.” Write the logic expression for the system and construct the truth table.
7. An access control panel grants entry only if both a valid keycard is scanned and the correct PIN is entered. Let [latex]K[/latex]: “Keycard is valid” and [latex]P[/latex]: “PIN is correct.” Write the logic expression and determine the output when [latex]K = 1[/latex] and [latex]P = 0[/latex].
8. An alarm system is triggered if the door is open and the security system is armed. Let [latex]D[/latex]: “Door is open” and [latex]S[/latex]: “System is armed.” Write the logic expression and evaluate the output when [latex]D = 1[/latex] and [latex]S = 1[/latex].
9. An automated fan turns on if the temperature is high or the humidity is high. Let [latex]T[/latex]: “Temperature is high” and [latex]H[/latex]: “Humidity is high.” Write the logic expression and determine the output when [latex]T = 0[/latex] and [latex]H = 1[/latex].
10. A data backup is initiated only if the system is idle and the backup schedule is active. Let [latex]I[/latex]: “System is idle” and [latex]B[/latex]: “Backup schedule is active.” Write the logic expression and evaluate the output when [latex]I = 1[/latex] and [latex]B = 0[/latex].

Challenge Problems

11. Design a combinatorial circuit that outputs 1 if at least two out of three inputs ([latex]A[/latex], [latex]B[/latex], [latex]C[/latex]) are 1. Write the Boolean expression and construct the truth table.
12. Given the Boolean expression [latex]A \cdot (B + C) + A \cdot B[/latex], simplify the expression using Boolean algebra laws and describe the resulting circuit.
13. A system triggers a fault alert if either sensor [latex]A[/latex] or sensor [latex]B[/latex] detects an error, but not both at the same time. Write the Boolean expression and construct the truth table.
14. A circuit is defined by the expression [latex](A \cdot B) + (A \cdot B) \cdot C[/latex]. Simplify the expression and explain how the simplified version reduces gate usage.
15. Consider a circuit with inputs [latex]A[/latex], [latex]B[/latex], and [latex]C[/latex], and output defined by [latex]((A + B) \cdot C) + (A \cdot B)[/latex]. Evaluate the output for all combinations of [latex]A[/latex], [latex]B[/latex], and [latex]C[/latex], and identify which combinations produce an output of 1.

6.2 Boolean Algebras

Basic Skills

16. List the three fundamental operations in Boolean algebra and describe their symbolic representations.
17. Evaluate the Boolean expression [latex]A + (A \cdot B)[/latex] for the input values [latex]A = 0[/latex] and [latex]B = 1[/latex].
18. Simplify the expression [latex]A + 0[/latex] and state which law(s) you used.
19. Simplify the expression [latex]A \cdot \bar{A}[/latex] and state which law(s) you used.
20. Simplify the expression [latex]A \cdot (B + C)[/latex] and state which law(s) you used.

Applications

21. A digital lock opens only if the correct code is entered and the override switch is not active. Let [latex]C[/latex]: “Correct code entered” and [latex]O[/latex]: “Override switch is active.” Write the Boolean expression for the lock mechanism and evaluate it when [latex]C = 1[/latex] and [latex]O = 0[/latex].
22. An email is marked as spam if it contains suspicious keywords or is from an unverified sender. Let [latex]K[/latex]: “Contains suspicious keywords” and [latex]U[/latex]: “Sender is unverified.” Write the Boolean expression and construct the truth table.
23. A server maintenance alert is triggered if the server is offline or the temperature exceeds safe limits. Let [latex]S[/latex]: “Server is online” and [latex]T[/latex]: “Temperature is safe.” Write the Boolean expression using complements and evaluate the output when [latex]S = 0[/latex] and [latex]T = 1[/latex].
24. Access to a secure folder is granted only if the user is authenticated and not flagged as suspicious. Let [latex]A[/latex]: “User is authenticated” and [latex]F[/latex]: “User is flagged.” Write the Boolean expression and determine the output when [latex]A = 1[/latex] and [latex]F = 0[/latex].
25. A backup power system activates if the main power fails or the battery level is critically low. Let [latex]P[/latex]: “Main power is active” and [latex]B[/latex]: “Battery level is sufficient.” Write the Boolean expression using complements and evaluate the output when [latex]P = 0[/latex] and [latex]B = 1[/latex].

Challenge Problems

26. Simplify the Boolean expression [latex](A + B) \cdot (A + \bar{B})[/latex] using Boolean algebra laws. Show each step and identify which laws are applied.
27. Prove the Boolean identity [latex]A + (A \cdot B) = A[/latex] using a truth table and Boolean algebra laws.
28. Design a Boolean expression that outputs 1 only when exactly one of the inputs [latex]A[/latex], [latex]B[/latex], or [latex]C[/latex] is 1. Construct the truth table and simplify the expression.
29. Simplify the expression [latex]\bar{A \cdot B + C}[/latex] and state which law(s) you used. Then construct the truth table for both the original and simplified expressions to verify equivalence.
30. Prove the distributive law of Boolean algebra [latex]A \cdot (B + C) = (A \cdot B) + (A \cdot C)[/latex] using both a truth table and algebraic reasoning.

6.3 Boolean Functions and Synthesis of Circuits

Basic Skills

31. Write a Boolean function [latex]F(A, B)[/latex] that outputs 1 only when both inputs [latex]A[/latex] and [latex]B[/latex] are 1.
32. Construct the truth table for the Boolean function [latex]F(A, B, C) = (A \cdot B) + C[/latex]
33. Evaluate the Boolean function [latex]F(A, B) = A + \bar{B}[/latex] for the input values [latex]A = 0[/latex] and [latex]B = 1[/latex].
34. Given the Boolean function [latex]F(A, B) = A \oplus B[/latex] (exclusive OR), what is the output when [latex]A = 1[/latex] and [latex]B = 1[/latex]?
35. Simplify the Boolean expression [latex]F(A, B) = A \cdot (A + B)[/latex] using Boolean algebra laws.

Applications

36. A security system activates an alarm if motion is detected and the system is armed, or if a window sensor is triggered. Let [latex]M[/latex]: “Motion detected,” [latex]A[/latex]: “System armed,” and [latex]W[/latex]: “Window sensor triggered.” Write the Boolean function and construct the truth table.
37. A login system grants access if the username is correct and either the password is correct or a biometric scan is successful. Let [latex]U[/latex]: “Username correct,” [latex]P[/latex]: “Password correct,” and [latex]B[/latex]: “Biometric scan successful.” Write the Boolean function and evaluate it for [latex]U = 1[/latex], [latex]P = 0[/latex], [latex]B = 1[/latex].
38. An irrigation system activates if the soil is dry and it is not raining, or if the manual override is enabled. Let [latex]D[/latex]: “Soil is dry,” [latex]R[/latex]: “It is raining,” and [latex]O[/latex]: “Manual override enabled.” Write the Boolean function and determine the output when [latex]D = 1[/latex], [latex]R = 1[/latex], [latex]O = 0[/latex].
39. A data packet is sent if the network is available and the buffer is not full. Let [latex]N[/latex]: “Network available” and [latex]B[/latex]: “Buffer full.” Write the Boolean function and evaluate it when [latex]N = 1[/latex] and [latex]B = 0[/latex].
40. Design a Boolean function that outputs 1 only when inputs [latex]A[/latex] and [latex]B[/latex] are different. Construct the truth table and write the simplified Boolean expression.

Challenge Problems

41. Simplify the Boolean function [latex]F(A, B, C) = A \cdot B + A \cdot C + B \cdot C[/latex] using Boolean algebra laws. Show each step and explain how the simplification reduces the number of gates in a circuit.
42. Design a Boolean function [latex]F(A, B, C)[/latex] that outputs 1 if at least two of the three inputs are 1. Write the truth table and derive a simplified Boolean expression.
43. Construct a Boolean function for exclusive OR using only AND, OR, and NOT gates. Write the expression and explain how it behaves for all input combinations.
44. Given the Boolean function [latex]F(A, B, C) = (A \cdot \bar{B}) + (B \cdot C)[/latex], draw the logic gate structure and explain how the circuit processes inputs to produce the output.
45. A system has three status flags: [latex]A[/latex], [latex]B[/latex], and [latex]C[/latex]. The system is in an invalid state if all three flags are the same (either all 0 or all 1). Design a Boolean function that outputs 1 when the system is in an invalid state. Construct the truth table and simplify the expression.