CombinationalLogic 1

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Topics

• Basics of digital logic • Basic functions♦ Boolean algebra♦ Gates to implement Boolean functions

• Identities and Simplification

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Binary Logic

• Binary variables♦ Can be 0 or 1 (T or F, low or high)♦ Variables named with single letters in

examples♦ Use words when designing circuits

• Basic Functions♦ AND♦ OR♦ NOT

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AND Operator

• Symbol is dot♦ Z = X · Y

• Or no symbol♦ Z = XY

• Truth table ->• Z is 1 only if♦ Both X and Y are 1

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OR Operator

• Symbol is +♦ Not addition♦ Z = X + Y

• Truth table ->• Z is 1 if either 1♦ Or both!

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NOT Operator

• Unary• Symbol is bar (or ’)♦ Z = X’

• Truth table ->• Inversion

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Gates

• Circuit diagrams are traditionally used to document circuits

• Remember that 0 and 1 are represented by voltages

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AND Gate

Timing Diagrams

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OR Gate

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Inverter

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More Inputs

• Work same way• What’s output?

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Digital Circuit Representation: Schematic

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Digital Circuit Representation: Boolean Algebra

• For now equations with operators AND, OR, and NOT

• Can evaluate terms, then final OR

• Alternate representations next

ZY X F

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Digital Circuit Representation: Truth Table

• 2n rowswhere n # ofvariables

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Functions

• Can get same truth table with different functions

• Usually want simplest function♦ Fewest gates or using particular types

of gates♦ More on this later

ZY X F ))(( ZXY X F

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Identities

• Use identities to manipulate functions

• On previous slide, I used distributive law

to transform fromZY X F ))(( ZXY X F

))(( ZX YX YZ X

to

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Table of Identities

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Duals

• Left and right columns are duals

• Replace AND with OR, 0s with 1s

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Single Variable Identities

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Commutative

• Order independent

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Associative

• Independent of order in which we group

• So can also be written asand

ZYX XYZ

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Distributive

• Can substitute arbitrarily large algebraic expressions for the variables

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DeMorgan’s Theorem

• Used a lot• NOR equals invert AND

• NAND equals invert OR

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Truth Tables for DeMorgan’s

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Algebraic Manipulation

• Consider functionXZZYX YZ X F

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Simplify Function

XZZ Z YX F )(

XZYX F 1

XZYX F

Apply

Apply

Apply

XZZYX YZ X F

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Fewer GatesXZYX F

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Consensus Theorem

• The third term is redundant♦ Can just drop

• Proof in book, but in summary♦ For third term to be true, Y & Z both 1♦ Then one of the first two terms must be

1!

ZXXYYZZXXY

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Complement of a Function

• Definition:

1s & 0s swapped in truth table

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Truth Table of the Complement of a Function

X Y Z F = X + Y’Z F’

0 0 0 0 1

0 0 1 1 0

0 1 0 0 1

0 1 1 0 1

1 0 0 1 0

1 0 1 1 0

1 1 0 1 0

1 1 1 1 0

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Algebraic Form for Complement• Mechanical way to derive

algebraic form for the complement of a function

1. Take the dual• Recall: Interchange AND & OR, and 1s & 0s

2. Complement each literal (a literal is a variable complemented or not; e.g. x , x’ , y, y’ each is a literal)

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Example: Algebraic form for the complement of a function

F = X + Y’Z• To get the complement F’

1. Take dual of right hand side

X . (Y’ + Z)2. Complement each literal: X’ . (Y

+ Z’)

F’ = X’ . (Y + Z’)

Mechanically Go From Truth Table to Function

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From Truth Table to Function

• Consider a truth table• Can implement F

by taking OR of all terms that correspond to rows for which F is 1 “Standard Form” of

the function

XYZZXYZYXZY XZ YX F

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Standard Forms

• Not necessarily simplest F• But it’s mechanical way to go

from truth table to function

• Definitions:♦ Product terms – AND ĀBZ♦ Sum terms – OR X + Ā♦ This is logical product and sum, not

arithmetic

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Definition: Minterm

• Product term in which all variables appear once (complemented or not)

• For the variables X, Y and Z example minterms : X’Y’Z’, X’Y’Z, X’YZ’, …., XYZ

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Definition: Minterm (continued)

MinTerm

Each minterm represents exactly one combination of the binary variables in a truth table.

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Truth Tables of Minterms

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Number of Minterms

• For n variables, there will be 2n minterms

• Minterms are labeled from minterm 0, to minterm 2n-1

♦m0 , m1 , m2 , … , m2n

-2 , m2n

-1

• For n = 3, we have

♦m0 , m1 , m2 , m3 , m4 , m5 , m6 , m7

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Definition: Maxterm

• Sum term in which all variables appear once (complemented or not)

• For the variables X, Y and Z the maxterms are:

X+Y+Z , X+Y+Z’ …. , X’+Y’+Z’

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Definition: Maxterms (continued)

mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm,mxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx,mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm

Maxterm

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Truth Tables of Maxterms

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Minterm related to Maxterm

• Minterms and maxterms with same subscripts are complements

• Example33 MZYXYZXm

Mjm j

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Standard Form of F:Sum of Minterms

• OR all of the minterms of truth table for which the functionvalue is 1

• F = m0 + m2 + m5 + m7

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Complement of F

• Not surprisingly, just sum of the other minterms

• In this caseF’ = m1 + m3 + m4 + m6

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Product of Maxterms

• Recall that maxterm is true except for its own row

• So M1 is only false for 001

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Product of Maxterms

• F = m0 + m2 + m5 + m7

• Remember:♦ M1 is only false for 001♦ M3 is only false for 011♦ M4 is only false for 100♦ M6 is only false for 110

• Can express F as AND of M1, M3, M4, M6

6431 MMMMF

))(( ZYXZYXF ))(( ZYXZYX

or

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Recap

• Working (so far) with AND, OR, and NOT

• Algebraic identities• Algebraic simplification• Minterms and maxterms• Can now synthesize function

from truth table