## SIMPLIFICATION OF BOOLEAN FUNCTIONS

This experiment demonstrates the relationship between a Boolean function and the corresponding logic diagram. The Boolean functions are simplified using the map method, as discussed in Chapter 3. The logic diagrams are to be drawn using NAND gates, as explained in Section 3.7 (NAND and NOR Implementation). The IC's to be used are the following NAND gates:

7400 2-input NAND

7404 Inverter (1-input NAND)

7410 3-input NAND

7420 4-input NAND

If an input to a NAND gate is not used, it should not be left open, but, instead, should be connected to another input that is used. For example, if the circuit needs an inverter and there is an extra two-input gate available in a 7400 IC, then both inputs of the gate are to be connected together to form a single input for an inverter.

### Logic Diagram

This part of the experiment starts with a given logic diagram from which one proceeds to apply simplification procedures to reduce the number of gates and possibly the number of IC's. The logic diagram shown in Fig. 1 requires two IC's, a 7400 and a 7410. Note that the inverters for inputs x, y, and z are obtained from the remaining three gates in the 7400 IC. If the inverters were taken from a 7404 IC, the circuit would have required three IC's. Also note that in drawing SSI (Small-Scale Integration) circuits, the gates are not enclosed in blocks as done with MSI (Medium-Scale Integration) circuits.

Assign pin numbers to all inputs and outputs of the gates and connect the circuit with the x, y, and z inputs going to three switches and the output F to an indicator LED. Test the circuit by obtaining its truth table.

Obtain the Boolean function of the circuit and simplify it using the map method. Construct the simplified circuit without disconnecting the original circuit. Test both circuits by applying identical inputs to both and observing the separate outputs. Show that for each of the eight possible input combinations, the two circuits have identical outputs. This will prove that the simplified circuit behaves exactly as the original circuit.

### Boolean Functions

Given the two Boolean functions in sum of minterms:

F1 (A, B, C, D) = (0,1,4,5,8,9,10,12,13)

F2 (A, B, C, D) = (3,5,7,8,10,11,13,15)

Simplify the two functions by means of maps. Obtain a composite logic diagram with four inputs, A, B, C, and D, and two outputs, Fl and F2. Implement the two functions together using a minimum number of NAND ICs. Do not duplicate the same gate if the corresponding term is needed for both functions. Use any extra gates in existing IC's for inverters when possible. Connect the circuit and check its operation. The truth table for Fl and F2 obtained from the circuit should conform to the minterms listed.

### Complement

Plot the following Boolean function in a map:

F = A'D + BD + B'C + AB'D

Combine the 1's in the map to obtain the simplified function for F in sum of products. Then combine the 0's in the map to obtain the simplified function for F' also in sum of products. Implement both F and F' using NAND gates and connect the two circuits to the same input switches, but to separate output indicator LED's. Obtain the truth table of each circuit in the laboratory and show that they are the complements of each other.