1. Picture a typical 7-line LED display (e.g., one digit of a typical digital alarm clock --- see the diagram below). You are going to design a combinatorial circuit that controls whether the bottom-left LED is lit (LED z_₅ to be precise). LED z_₅ is lit for numbers 0, 2, 6, and 8; and blank for all other inputs --- including invalid inputs.
   1. Write the truth table for this circuit. (The circuit is for LED z_₅ only. Do not include columns for other LEDs.)
   2. Express the truth table in Sum-of-Products form.
   3. Draw the logic diagram for this combinatorial circuit in a "PLA-style".
2. (Harris & Harris exercise 2.24) Write Boolean expressions for the circuit shown below. You need not minimize the equations.
   
3. Draw the logic diagram corresponding to this boolean expression: (AB xor (B + C̅)) + A̅C. (There is a bar over the first "C" and the last "A".) You may use XOR gate.
4. Draw a logic diagram to show how to build an exclusive-or gate out of AND, OR, and NOT gates.
5. (From Null 3rd ed., Chapter 3, number 35.) Construct a truth table to describe the operation of this circuit:
   
6. Write a Boolean expression describing the circuit shown above. Your expression should describe the circuit as drawn (i.e., don't simplify it).
7. Now write a simplified Boolean expression based on the truth table.
8. (From Null, Chapter 3, number 42.) Little Susie is building a dog-training robots she calls CESAR (Canine Education Special Assistant Robot). Currently, she is working on the robot unit that will award treats when training puppies. First she needs to figure out when a puppy should get dog biscuit as a reward; then she can build the appropriate circuit for the robot. She has concluded the following:
   • Give the puppy a biscuit if it sits and wiggles but does not bark.
   • Give the puppy a biscuit if it bars an wiggles but does not sit.
   • Give the puppy a biscuit if it sits but does not wiggle or bark.
   • Give the puppy a biscuit if it sits, wiggles and barks.
   • Don't give the puppy a biscuit otherwise.
   Assume the following variables and values:
   • S: Sit (0 for not sitting; 1 for sitting)
   • W: Wiggles (0 for not wiggling; 1 for wiggling)
   • B: Barking (0 for not barking; 1 for barking)
   • F: Biscuit function (0, don't give the puppy a biscuit 1, give the poppy a biscuit)
   Construct a truth table, find a minimized Boolean function to implement the logic, then draw the logic diagram.
9. Design a circuit that returns true if a 5-bit input represents the integer 12 and false otherwise.
10. Design a circuit that takes two n-bit inputs and returns true if the two inputs are identical and false otherwise.
11. Show how to build a NAND gate using only NOR gates. Hint: Use De Morgan's Theorems. You can also use google to find more extensive discussions on the web.
12. (Satisfies Learning Objective FC2) Write a formal proof that {NOR} is functionally complete. Your proof must include sentences. Diagrams alone are not sufficient.

Updated Sunday, 9 January 2022, 4:29 PM