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Ch. 4 Boolean Algebra and Logic Gates

Boolean Algebra Basic Identities and DeMorgan’s TheoremBoolean Algebraic Equations

Sum of ProductProduct of Sum

Logic GatesAND, OR, NOTNAND, NOR, XOR, XNOR

Other Basic ConceptsPropagation Delay Time, Power Dissipation, Noise MarginFan In / Fan Out

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Boolean Algebra 논리 연산을 다루는 수학적인 논리 이론 (영국 수학자 George Boole, 1854년)Basic Identities of Boolean Algebra : Table 4-3

Commutative Law (교환법칙)Associative Law (결합법칙)Distributive Law (배분법칙)DeMorgan’s Theorem

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Multiplication by Boolean Algebra : AND Operation 0 · 0 = ?0 · 1 = ?1 · 0 = ?1 · 1 = ?

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AND OperationCheck : X = A · B

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Addition by Boolean Algebra : OR Operation 0 + 0 = ?0 + 1 = ?1 + 0 = ?1 + 1 = ?

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OR OperationCheck : X = A + B

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Inverter : NOT Operation

0 ?1 ?

?X A

=

=

=

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Example 4.1, p57

( )Z Y X Y= +

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Example 4.2, p58DeMorgan’s Theorem ?

?

X Y X Y

X Y X Y

+ =

= +

i

i

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Example 4.3, p59

( ) ( ) ?

(1 )

(1 )

X Y X Z X YZXX XZ XY YZX XZ XY YZX XY XZ YZX Y XZ YZX XZ YZX Z YZX YZ

+ + = +

= + + += + + += + + += + + += + += + += +

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Example 4.4, p60

?XY YZ XZ XY YZ+ + = +

(1 ) (1 )

( )

XY YZ XY Z YZ XXY XYZ YZ YZXXY YZ XZY XZYXY YZ XZ Y YXY YZ XZ

+ = + + +

= + + +

= + + +

= + + +

= + +

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Duality : Table 4-4Boolean Function

0 11 0OR (+) AND (·)AND (·) OR (+)Logic variable : No change

1 1 0 0X X+ = → =i

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Inverting of a Boolean Function0 11 0OR (+) AND (·)AND (·) OR (+)Logic variable Inverting

(ex.1) (ex.2) p61

1 100

X XX

+ =

==

ii

Inverting of 1 X+

( ) ( )( ) ( )

1 1

1

0

X Y Z X Y Z

X Y Z

X Y Z

+ =

= + +

= + +

i i i i i

i

i

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(Ex. 4.5) p61

(Ex. 4.6) p62

?WX YZ+ =

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

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Boolean Algebraic Functions (or Equations)Truth TableSum of ProductProduct of Sum

Truth Table(Ex. 4.7) p63

Assign variables for three doors : A, B, CState 1 : Turn on the lightState 0 : Turn off the lightTotal 8 ( ) casesTurn on the light when at least two doors have been closed.

32=

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Truth Table for Ex. 4.7 : Table 4-6

Input Output

0 0 0 0 0

1 0 0 1 0

2 0 1 0 0

3 0 1 1 1

4 1 0 0 0

5 1 0 1 1

6 1 1 0 1

17 1 1 1

Cases

CA B D

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Boolean Algebraic Function by the Sum of Product FormInput Output Minterms

0 0 0 0 0

1 0 0 1 0

2 0 1 0 0

3 0 1 1 1

4 1 0 0 0

5 1 0 1 1

6 1 1 0 1

17 1 1 1

Cases

CA B DABCABCABCABCABCABCABCABC

D ABC ABC ABC ABC= + + +

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Simplification of Boolean FunctionsUsing Table 4-3

(Ex. 4.8) p64~65Truth Table : Table 4-8

( ) ( ) ( )( ) ( ) ( )

( )

D ABC ABC ABC ABCABC ABC ABC ABC ABC ABC X X XA A BC AC B B AB C C

BC AC ABC A B AB

= + + +

= + + + + + ← + =

= + + + + += + += + +

( ) ( )

( )

A XYZ XYZ XYZ XYZXZ Y Y XZ Y YXZ XZX X Z

Z

= + + +

= + + +

= +

= +

=

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Boolean Algebraic Function by the Product of Sum FormInput Output Maxterms

0 0 0 0 0

1 0 0 1 0

2 0 1 0 0

3 0 1 1 1

4 1 0 0 0

5 1 0 1 1

6 1 1 0 1

17 1 1 1

Cases

CA B DA B C+ +A B C+ +

A B C+ +A B C+ +A B C+ +A B C+ +

A B C+ +A B C+ +

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

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Simplification of Boolean FunctionsUsing Table 4-3

( )( )( )( )

( )( )( )( )

( ) ( ) ( ) ( )

( )

D A B C A B C A B C A B C

D A B C A B C A B C A B C

A B C A B C A B C A B C

ABC ABC ABC ABCABC ABC ABC ABCAB C C ABC ABCAB ABC ABC

= + + + + + + + +

= + + + + + + + +

= + + + + + + + + + + +

= + + +

= + + +

= + + +

= + +

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( ) , by Sum of ProductD C A B AB= + +∵

( )( )( )( )( )( )( ) (1 )( )( ) (1 )( )( )

( 1)

( )(

D AB ABC ABCA B A B C A B CAA AB AC BA BB BC A B CA AB AC AB BC A B C A B B AA AC BC A B C A C AA BC A B C

AA AB AC BCA BCB BCCAB AC BCA BC BC A BCAB AC BCAB A B CC

= + +

= + + + + +

= + + + + + + +

= + + + + + + ← + + =

= + + + + ← + =

= + + +

= + + + + +

= + + + ← + == + += + += )A B AB+ +

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Logic Gates하나 또는 그 이상의 입력신호가 출력신호를 야기하도록 작동하는 전자회로

AND, OR, NOT, NAND, NOR, XOR, XNOR

AND, OR, NOT gates

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Graphic Symbols of AND, OR, NOT gates

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Multi-Input Symbols of AND, OR gates

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Electrical Analogy of AND, OR, NOT gatesTR (Transistor) as a Switching DeviceResistor

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IC (Integrated Circuit) Chips – AND gates

74LS08 74HC11 74HC21

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Physical View of IC (Integrated Circuit) Chips

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Applications of AND gates

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Applications of AND gates

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Implementation and Evaluation by Experiments

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IC (Integrated Circuit) Chips – OR, NOT gates

74LS32 7404 4049

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Applications of OR gates

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Other Gates, p69NANDNORXORXNOR

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NAND and NOR Gates

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Applications of NAND gateNOT gate operation

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Applications of NAND gateAND gate operation

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Applications of NAND gateOR gate operation

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Applications of NAND gateWith a Control input

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Applications of NOR gate

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Applications of XOR gateUseful Identities

Parity Generation and Checking

0 10 1

( ) ( )

X X X XX X X X

X Y X Y X Y X YA B B AA B C A B C

⊕ = ⊕ =

⊕ = ⊕ =

⊕ = ⊕ ⊕ = ⊕⊕ = ⊕⊕ ⊕ = ⊕ ⊕

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Other Basic ConceptsPropagation Delay TimePower DissipationNoise MarginFan In / Fan Out

Propagation Delay Time Fig. 4-15Dependant on the serial connections of logic gates

( )D ABC ABC ABC ABC

C A B AB= + + += + +

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Power Dissipation(Ex. 4.10) Average power dissipation

Noise MarginFig. 4-16

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AVR CC CC

CCH CCLCC

P V I

I II n

=

+⎛ ⎞= ⎜ ⎟⎝ ⎠

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Fan In / Fan Out

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Homework, Pages 83~841-(2)23-(4)678