 # Boolean Algebra and Logic Simplification

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Boolean Algebra and Logic Simplification
Chapter 4 Boolean Algebra and Logic Simplification

Boolean Operations and Expressions

Boolean Multiplication (AND)

Laws and Rules of Boolean Algebra
Commutative Associative Distributive Rules

Figure 4--1 Application of commutative law of addition.

Figure 4--2 Application of commutative law of multiplication.

Figure 4--3 Application of associative law of addition

Figure 4--4 Application of associative law of multiplication

Figure 4--5 Application of distributive law

Rules of Boolean Algebra

Figure RULE 1

Figure RULE 2

Figure RULE 3

Figure RULE 4

Figure RULE 5

Figure RULE 6

Figure RULE 7

Figure RULE 8

Figure RULE 9

DeMorgan’s Theorem Figure Gate equivalencies and the corresponding truth tables that illustrate DeMorgan’s theorems. Notice the equality of the two output columns in each table. This shows that the equivalent gates perform the same logic function.

Boolean Expression for a Logic Circuit
Figure A logic circuit showing the development of the Boolean expression for the output.

Constructing a Truth Table for a Logic Circuit
Evaluating the expression and putting results in truth table format

Simplification Using Boolean Algebra

Figure 4--17 Gate circuits for Example 4-8

Standard Forms of Boolean Expressions
Sum-of-Products (SOP) Form Product-of-Sum (POS) Form

Figure 4--18 Implementation of the SOP expression AB + BCD + AC.
SOP Form Figure Implementation of the SOP expression AB + BCD + AC.

Standard SOP Form

Binary Representation of Product Term

POS Form Figure Implementation of the POS expression (A + B)(B + C + D) (A + C).

Standard POS Form

Binary Representation of Sum Term

Converting Standard SOP to Standard POS

Converting SOP to Truth Table

Converting POS to Truth Table

Determining Standard Expressions from Truth Table

Figure 4--20 A 3-variable Karnaugh map showing product terms.

Figure 4--21 A 4-variable Karnaugh map.

Figure Adjacent cells on a Karnaugh map are those that differ by only one variable. Arrows point between adjacent cells.

Figure 4--23 Example of mapping a standard SOP expression.
Karnaugh Map SOP Minimization Figure Example of mapping a standard SOP expression.

Figure 4--24

Figure 4--25

Mapping Nonstandard SOP Expression

Figure 4--26

Figure 4--27

Karnough Map Simplification of SOP Expressions
Example Group the 1s in each Karnaugh maps

Figure 4--29

Figure Determine SOP Related Problem: add 1 in the lower right cell (1010) and determine the resulting SOP.

Figure 4--31 Related Problem: For the Karnaugh map in Fig. 4-31(d), add 1 in the 0111 cell and determine the resulting SOP.

Mapping Directly from Truth Table
Figure Example of mapping directly from a truth table to a Karnaugh map.

“Don’t Care” Conditions
Figure Example of the use of “don’t care” conditions to simplify an expression.

Karnaugh Map POS Minimization
Figure Example of mapping a standard POS expression.

Example 4-30

Figure 4--38

Figure 4--39

Converting b/w POS and SOP Using Karnaugh Map

Five-Variable Karnaugh Maps
Figure A 5-variable Karnaugh map.

Figure 4--42 Illustration of groupings of 1s in adjacent cells of a 5-variable map.

Figure 4--43

Figure 4--44 Basic structure of a PAL.
Programmable Logic Figure Basic structure of a PAL.

Figure 4--45 PAL implementation of a sum-of-products expression.

Digital System Application
Figure Seven-segment display format showing arrangement of segments.

Figure 4--52 Display of decimal digits with a 7-segment device.

Figure 4--53 Arrangements of 7-segment LED displays.

Figure 4--55 Block diagram of 7-segment logic and display.

Figure 4--56 Karnaugh map minimization of the segment-a logic expression.

Figure 4--57 The minimum logic implementation for segment a of the 7-segment display.