Activity 4.2B – DeMorgan’s Theorems

Purpose 1. To understand the DeMorgan's Theorems. 2. To apply DeMorgan's Theorems to complex Boolean expressions to simplify them 3. To simulate original and simplified circuits to verify equivalency of truth tables.

Equipment Computer Simulation software

Procedure DeMorgan's theorems are important tools in Boolean Algebra. DeMorgan's theorems provide a mathematical verification of the equivalency of: 1. NAND gates and negative - OR gates 2. NOR gates and negative - AND gates

A negative - OR gate is an OR gate with inverters on its inputs. They can be represented by the drawings below:

negative - OR

A negative input AND gate is an AND gate with inverters on its inputs. They can be represented by the drawings below:

negative - AND

The two DeMorgan's Theorems can be stated as follows:

The complement of a product of variables is equal to the sum of the complements of the variables.

The complement of a sum of variables is equal to the product of the complements of the variables.

The gates drawn below show the equivalencies for the two theorems:

When properly used the theorems allow you to remove inverter bars over two or more variables in the expression. The final simplification will be an expression reduced to single bars over single terms. Example:

Apply DeMorgan's theorems to simplify the expression:

Solution:

Let: A+B+C = X.

Then:

Apply the first DeMorgan's Theorem - the expression becomes:

Now, apply DeMorgan's Theorem to the term:

The simplified expression has no bars over more than one term.

1. Using software simulation, build the following expression:

Tape the printout of the circuit below:

2. Fill in the Truth Table below:

3. Simplify the circuit using DeMorgan's Theorems: (show math )

4. Using the simulation software, build the simplified circuit. Tape a printout of simplified circuit below:

5. Fill in the Truth Table below:

6. Simulate the following circuit and test its operation. Add logic switches to the circuit inputs and an LED to the output.

7. Fill in the Truth Table below:

Demonstrate the simulation to your teacher and get signature:______

8. Simplify the circuit using DeMorgan's Theorems: (show math )

9. Simulate the simplified circuit.

10. Fill in the Truth Table below:

Demonstrate the simulation to your teacher and get signature:______

11. Simplify the following expression in the space below. (Show all steps used.)

12. Construct the unsimplified and simplified circuit using the computer simulation software. Use logic switches and an output LED indicator. Tape the printouts of the both circuits below:

13. Test the unsimplified circuit and fill in the Truth Table below:

Test the simplified circuit. What do you notice about the output values of T?

14. Breadboard both the simplified and the unsimplified circuit and test for equivalency by comparing the output values obtained to the truth table above.

Demonstrate the circuits to your teacher and get signature:______