digital fundamentals

Digital Fundamentals, Eleventh EditionThomas L. Floyd

Digital Fundamentals

Boolean Algebra and Logic Simplification

FIGURE 4-1

FIGURE 4-2

FIGURE Application of commutative law of addition.

FIGURE Application of commutative law of multiplication.

FIGURE Application of associative law of addition.

FIGURE Application of associative law of multiplication.

FIGURE Application of distributive law.

• commutative law: A+B =?

• associative law: A +(B+C) =?

A(BC) =?

• distributive law: A(B+C)=?

(A+B)(C+D)=?

TABLE 4–1 Basic rules of Boolean algebra.

FIGURE 4-8

Anything ORed with 0 equals to itself

FIGURE 4-9

Anything ORed with 1 equals to 1

FIGURE 4-10

Anything ANDed with 0 equals to 0

FIGURE 4-11

Anything ANDed with 1 equals to

itself

FIGURE 4-12

Anything ORed with itself equals to

itself

FIGURE 4-13

Anything ORed with its own

complement equals to 1

FIGURE 4-14

Anything ANDed with its own

complement equals to itself

FIGURE 4-15

Anything ANDed with its own

complement equals to 0

FIGURE 4-16

Anything complemented twice will

return to its original logic level

TABLE 4–2 Rule 10: A + AB = A.

Elimination: One or more variables in

second term can be eliminated.

Used in simplication

TABLE 4–3 Rule 11: A + ĀB = A + B.

Absorption: One or more variables in second term can be

eliminated. Used in simplication

Ā+AB =Ā+B (?)

TABLE 4–4 Rule 12: (A + B)(A + C) = A + BC.

Absorption: Used in simplication

DEMORGAN'S THEOREMS

FIGURE 4-17 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.

FIGURE 4-18 A combinational logic circuit showing the development of the Boolean expression for the output.

FIGURE 4-20 Gate circuits

Simplify the following Boolean expressions :

STANDARD FORMS OF BOOLEAN EXPRESSIONS

FIGURE 4-22 Implementation of the SOP expression AB + BCD + AC.

FIGURE 4-23 This NAND/NAND implementation is equivalent to the AND/OR in Figure 4–22.

FIGURE 4-24 Implementation of the POS expression (A + B)(B + C + D)(A + C).

Circuit Simplification Using Boolean Algebra

Karnaugh Map

• K-Map: a tools to preform a systematic reduction of complex logic circuits into simplified equivalent circuit.

• Transform the Boolean equation to be reduced into SOP expression

• Fill in a proper cell of the K-map

• Encircle adjacent cells in groups of two, four, or eight

• Find each term of the final SOP equation by determining which variable remain constant within each circle.

FIGURE 4-25 A 3-variable Karnaugh map showing Boolean product terms for each cell.

FIGURE 4-26 A 4-variable Karnaugh map.

FIGURE 4–27 Adjacent cells on a Karnaugh map are those that differ by only one variable. Arrows point between adjacent cells.

FIGURE 4-28 Example of mapping a standard SOP expression.

Transform the Boolean equation to be reduced into SOP expression

Fill in a proper cell of the K-map

FIGURE 4-29

FIGURE 4-30

FIGURE 4-33

Encircle adjacent cells in groups of two, four, or eight ?

FIGURE 4-34

Encircle adjacent cells in groups of two, four, or eight

FIGURE 4-35

Find each term of the final SOP equation by determining which

variable remain constant within each circle.

FIGURE 4-36

Find each term of the final SOP equation by determining

which variable remain constant within each circle.

FIGURE 4-37

FIGURE 4-39 Example of mapping directly from a truth table to a Karnaugh map.

FIGURE 4-43 Example of mapping a standard POS expression.

digital fundamentals · 2016. 11. 24. · digital fundamentals, eleventh edition thomas l. floyd...

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