1. Boolean Algebra and Reduction Techniques
Chapter 5
Boolean Algebra and Reduction Techniques 1

2. Objectives You should be able to:
Write Boolean equations for combinational logic applications.
Use Boolean algebra laws and rules to simplify combinational logic circuits.
Apply DeMorgan’s theorem to complex Boolean equations to arrive at simplified equivalent equations. 2

3. Objectives (Continued)
Design single-gate logic circuits by using the universal capability of NAND and NOR gates.
Troubleshoot combinational logic circuits.
Implement sum-of-products expressions using AND-OR-INVERT gates. 3

4. Objectives (Continued)
Use the Karnaugh mapping procedure to systematically reduce complex Boolean equations to their simplest form.
Describe the steps involved in solving a complete system design application. 4

5. Combinational Logic
Using two or more logic gates to perform a more useful, complex function
A combination of logic functions
B = KD + HD
Boolean Reduction
B = D(K+H) 5

6. Discussion Point
Write the Boolean equation for the circuit below: 6

7. VHDL Proof A circuit reduction can easily be proved using VHDL
Write a program for each equation
Run a simulation for all possible input conditions
Compare the results

8. Boolean Laws and Rules
Commutative laws of addition and multiplication: The order of the variables does not matter.
A + B = B + A
AB = BA 7

9. Boolean Laws and Rules Associative laws of addition and multiplication
A + (B + C) = (A + B) + C
A(BC) = (AB)C 8

10. Boolean Laws and Rules Distributive laws A(B + C) = AB + AC
(A + B)(C + D) = AC + AD + BC + BD 9

11. Boolean Laws and Rules Rule 1: Anything ANDed with a 0 equals 0
Rule 2: Anything ANDed with a 1 equals itself
A • 1 = A 10

12. Boolean Laws and Rules Rule 3: Anything ORed with a 0 equals itself
A + 0 = A
Rule 4: Anything ORed with a 1 is equal to 1
A + 1 = 1 11

13. Boolean Laws and Rules
Rule 5: Anything ANDed with itself is equal to itself
A • A = A
Rule 6: Anything ORed with itself is equal to itself
A + A = A 12

14. Boolean Laws and Rules
Rule 7: Anything ANDed with its complement equals 0
A • A = 0
Rule 8: Anything ORed with its complement equals 1
A + A = 1 13

15. Boolean Laws and Rules
Rule 9: Anything complemented twice will return to its original logic level
A = A 14

16. Boolean Laws and Rules
Rule 10:
A + Ā B = A + B
Ā + AB = Ā + B 15

17. 16

18. Discussion Point Which Boolean laws are illustrated below?
B + (D + E) = (B + D) + E
AB = BA
A + B + C = B + C + A
A(C + D) = AC + AD
What are some strategies for remembering the 10 Boolean rules? 17

19. Simplification of Combinational Logic Circuits Using Boolean Algebra
Equivalent circuits can be formed with fewer gates
Cost is reduced
Reliability is improved
Use laws and rules of Boolean Algebra 18

20. Simplification of Combinational Logic Circuits Using Boolean Algebra
Simplify the logic circuit shown by using the appropriate laws and rules. 19

21. Simplification of Combinational Logic Circuits Using Boolean Algebra
Simplify the logic circuit shown by using the appropriate laws and rules. 20

22. Using Quartus II to Simplify Equations
Quartus II software determines the simplest form of an equation during compiling.
It eliminates unnecessary gates
Minimizes the number of gates in the FPGA

23. DeMorgan’s Theorem
Used to simplify circuits containing NAND and NOR gates
A B = A + B
A + B = A B 21

24. DeMorgan’s Theorem
Break the bar over the variables and change the sign between them
Inversion bubbles - used instead of inverters to show inversion.
Use parentheses to maintain proper groupings
Results in Sum-of-Products (SOP) form 22

25. DeMorgan’s Theorem
Bubble Pushing 23

26. DeMorgan’s Theorem Bubble Pushing Change the logic gate
(AND to OR or OR to AND)
Add bubbles to the inputs and outputs where there were none and remove original bubbles 24

27. Truth Tables in VHDL using Vector Signals
Define inputs as an internal signal
Inputs are grouped as vectors
Assign values to the elements of the vector
Assign outputs for each input combination

28. The Universal Capability of NAND and NOR Gates
The NAND as an inverter. 25

29. The Universal Capability of NAND and NOR Gates
Forming an AND with two NANDs 26

30. The Universal Capability of NAND and NOR Gates
Forming an OR with three NANDs 27

31. The Universal Capability of NAND and NOR Gates
Forming a NOR with four NANDs 28

32. Discussion Point
The technique used to form all gates from NANDs can also be used with NOR gates.
Here is an inverter:
Form the other logic gates using only NORs. 29

33. AND-OR-INVERT Gates for Implementing Sum-of-Products Expressions
Product-of-sums (POS) form 30

34. AND-OR-INVERT Gates for Implementing Sum-of-Products Expressions
Sum-of-products (SOP) form 30

35. Karnaugh Mapping Used to minimize the number of gates
Reduce circuit cost
Reduce physical size
Reduce gate failures
Requires SOP form 31

36. Karnaugh Mapping
Graphically shows output level for all possible input combinations
Moving from one cell to an adjacent cell, only one variable changes 32

37. Karnaugh Mapping Steps for K-map reduction:
Transform the Boolean equation into SOP form
Fill in the appropriate cells of the K-map
Encircle adjacent cells in groups of 2, 4 or 8
Watch for the wraparound
Find terms by determining which variables remain constant within circles 33

38. Discussion Point
Use a K-map to simplify the circuit. 34

39. System Design Applications
Use Karnaugh Mapping to reduce equations
Use AND-OR-INVERT gates to implement logic 35

40. System Design Applications
Use K-maps to simplify less complex circuits before implementing the circuit using an AOI.
More complex circuits are better suited for implementation using PLDs. 36

41. Summary
Several logic gates can be connected together to form combinational logic.
There are several Boolean laws and rules that provide the means to form equivalent circuits.
Boolean algebra is used to reduce logic circuits to simpler equivalent circuits that function identically to the original circuit. 41

42. Summary
DeMorgan’s theorem is required in the reduction process whenever inversion bars cover more than one variable in the original Boolean equation.
NAND and NOR gates are sometimes referred to as universal gates, because they can be used to form any of the other gates. 42

43. Summary
AND-OR-INVERT (AOI) gates are often used to implement sum-of-products (SOP) equations.
Karnaugh mapping provides a systematic method of reducing logic circuits.
Combinational logic designs can be entered into a computer using schematic block design software or VHDL. 43

44. Summary
Using vectors in VHDL is a convenient way to group like signals together similar to an array.
Truth tables can be implemented in VHDL using vector signals with the selected signal assignment statement.
Quartus II can be used to determine the simplified equation of combinational circuits. 43