# Boolean Algebra and Reduction Techniques

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

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

Objectives You should be able to:
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

Objectives You should be able to:
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

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

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

Boolean Laws and Rules Commutative law of addition and multiplication
A + B = B + A ABC = BCA Figures 5-7 and 5-8 7

Boolean Laws and Rules Associative law of addition and multiplication
A + (B + C) = (A + B) + C A(BC) = (AB)C Figures 5-9 and 5-10 8

Boolean Laws and Rules Distributive law A(B + C) = AB + AC
(A + B)(C + D) = AC + AD + BC + BD Figures 5-11 and 5-12 9

Boolean Laws and Rules Rule 1: Anything ANDed with a 0 is equal to 0
Rule 2: Anything ANDed with a 1 is equal to itself Figure 5-14 10

Boolean Laws and Rules Rule 3: Anything ORed with a 0 is equal to itself Figure 5-15 Rule 4: Anything ORed with a 1 is equal to 1 Figure 5-16 11

Boolean Laws and Rules Rule 5: Anything ANDed with itself is equal to itself Figure 5-17 Rule 6: Anything ORed with itself is equal to itself Figure 5-18 12

Boolean Laws and Rules Rule 7: Anything ANDed with its own complement equals 0 Figure 5-19 Rule 8: Anything ORed with its own complement equals 1 Figure 5-20 13

Boolean Laws and Rules Rule 9: Anything complemented twice will return to its original logic level Figure 5-21 14

Boolean Laws and Rules Rule 10: A + AB = A + B
See Table 5-1 in your text 15

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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

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

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

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

DeMorgan’s Theorem To simplify circuits containing NAND and NOR gates
A B = A + B A + B = A B 21

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 Use of the MultiSIM logic converter 22

DeMorgan’s Theorem Bubble Pushing Figure 5-46 23

DeMorgan’s Theorem Bubble Pushing
shortcut method of forming equivalent gates 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

The Universal Capability of NAND and NOR Gates
The NAND as an inverter. Figure 5-49(a) 25

The Universal Capability of NAND and NOR Gates
Forming an AND with two NANDs Figure 5-49(b) 26

The Universal Capability of NAND and NOR Gates
Forming an OR with three NANDs Figure 5-53 27

The Universal Capability of NAND and NOR Gates
Forming a NOR with three NANDs Figure 5-54 28

Discussion Point The technique used to form all gates from NANDs can also be used with NOR gates. Here is an inverter – Figure 5-55 Form the other logic gates using only NORs. 29

AND-OR-INVERT Gates for Implementing Sum-of-Products Expressions
Product-of-sums (POS) form Sum-of-products (SOP) form Can easily be implemented using an AOI gate Programmable Logic Devices (PLDs) Can also be used 30

Karnaugh Mapping To minimize the number of gates Reduce circuit cost
Reduce physical size Reduce gate failures Requires SOP form 31

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

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

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

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

System Design Applications
Use a K-map to simplify a circuit that will use an AOI and inverters to output a HIGH when a 4 bit hexadecimal input is an odd number from 0 to 9 36

CPLD Design Applications
Used to simulate combinations of inputs and observe the resulting output to check for proper design operation. See CPLD Applications 5-1 and 5-2 37

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Figures 5-80 and 5-81 40

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

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

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. 43