03 boolean algebra digital logic
TRANSCRIPT
7/23/2019 03 Boolean Algebra Digital Logic
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Boolean Algebra and
Digital Logic
Chapter 7
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Symbols
• variables x, y, z
xy means x and y (logic used !
x " y means x and y (logic used !x # y means x or y (logic used v!
x means not x (logic used $ or ¬!
x% means not x (logic used $ or¬!
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&ruth &ables
And Or Not orInverter
x y x*y x y x+y x x'
' ' ' ' ' ' '
' ' ' '
' ' '
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Boolean La)s
• De *organ
• Associative
• Commutative
• Distributive ))((
)(
z x y x yz x
xz xy z y x
++=+
+=+
y x y x
y x y x
=+
+=⋅
z xy yz x
z y x z y x
)()(
)()(
=
++=++
yx xy
x y y x
=
+=+
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Boolean La)s
• +dentity
• Complement
• +dempotent
• Absorption
• Boundedness
•+nvolution
x x x
x x x
=⋅
=+ x x =
0
1
=⋅
=+
x x
x x
x x
x x
=⋅
=+
1
0
00
11
=⋅
=+
x
x
x y x x
x xy x
=+
=+
)(
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Booleaneductions
• Derived
)(
)(
)(0)(
)()(
)()(
)(
)(
)(
demorgan y x
bounded xy
complement xyvedistributi xy x x
involution y x x
demorgan y x x
involution y x x
demorgan y x x
involution y x x
y x x
+
+
+
+
+
+
+
y x y x x +=+
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Booleaneductions• Derived -
)(
)(
)(
)(0
)(
)()(
)()(
)(
)(
involution y x
demorgan y x
bounded y x
complement y x
distribut y x x x
involution y x x
demorgan y x x
demorgan y x x
involution y x x
y x x
+
+
+
+
+
+
+
+
y x y x x +=+
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Boolean &erms
• Boolean variable . or ' (yes or no, true or
/alse!
• Boolean expression 0 Boolean variables
connected )ith operators• 12uivalent Boolean 1xpressions 0 i/ every
possible combination o/ variable valuesevaluates to the same value3
•4roo/ by 4er/ect +nduction 0 – Construct truth table /or both expressions
– +/ both truth tables come to the same conclusionthen the t)o expressions are e2uivalent3
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&ruth 5unctions
• A truth function is a /unction o/ type6true,/alsex6true,/alsex333x6true,/alse.86true,/alse3
• &he number o/ sets in the product (o/the domain! depends on the number o/arguments that the /unction has3 (-n!
• An example o/ a truth /unction is y x y x y x f +=),(
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Boolean La)s
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)1(),(
)(),(
)(1),(
)()(),(
)()(),(
),( Reduce
derived y x y x f
identity y x x y x f
complement y x x y x f
distrib y x y y x y x f
commut y x y x y x y x f
y x y x y x y x f
+=
+=
+⋅=
++=
++=
++=
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)(),(
)(11),(
)()()(),(
)()()(),(
)()(),(
),( Reduce
identity y x y x f
complement y x y x f
distrib x x y y y x y x f
commut y x y x y x y x y x f
idempotent y x y x y x y x y x f
y x y x y x y x f
+=
⋅+⋅=
+++=
+++=
+++=
++=
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Boolean 4ractice
• educe y x y x xy y x f ++=),(
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&ruth &ables
• &urn this /unction into a truth table
x y x% y% x%y xy% x%y#xy%
' ' ' ' '
' ' ' ' ' '
' ' ' ' '
y x y x y x f +=),(
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&ruth &ables
5rom the truth table )e can see that )hen x is' and y is or x is and y is ' then /(x,y! is
x y /(x,y!
' ' '
'
'
'
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&ruth &ables
(x is ' and y is ! or (x is and y is '!/(x,y! is
ynotandor xyandnot x),( = y x f
y x y x y x f +=),(
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&ruth &ables
• See the pattern
• @here /(x,y! is a include the minterm in the
D?5 or S<4x y x% y% x%y xy% x%y#xy% *inter
m
' ' ' ' ' x%y%
' ' ' x%y ' ' ' x y%
' ' ' ' ' x y
y x y x y x f +=),(
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&ruth &ables
@rite the S<4 Boolean e2uation3
x y /(x,y!
' '
' '
' '
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&ruth &ables
@rite the S<4 Boolean e2uation3
x y /(x,y!
' '
' '
' '
xy y x y x f +=),(
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&ruth &ables
+s the /ollo)ing e2uation in its simplest /orm
x y /(x,y!
' '
' '
' '
xy y x y x f +=),(
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&ruth &ables
@rite the S<4 Boolean e2uation /rom the given truthtable
/(x,y!
x y /(x,y!
' ' '
'
'
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&ruth &ables
educe i/ possible
x y /(x,y!
' ' '
'
'
xy y x y x y x f ++=),(
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Create an S<4 /rom the given truthtable
x y z /(x,y,z!
' ' '
' ' ' '
'
' '
' '
' '
'
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Create an S<4 /rom the given truthtable
x y z /(x,y,z!
' ' '
' ' ' '
'
' '
' '
' '
'
z y x yz x z y x z y x z y x z y x f ++++=),,(
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educe using booleanalgebra
)(),,(
),,(
here)stop(Could ),,(
)(),,(
),,(
)()()(),,(
),,(
),,(
z y x z y x f
z y x z y x f
z y x z y x f
z y y y x z y x f
z y y x y x z y x f
x x z y z z y x z z y x z y x f
z y x z y x yz x z y x z y x z y x z y x f
z y x yz x z y x z y x z y x z y x f
+=
++=
+=
++=
++=
+++++=
+++++=
++++=
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@hat are the /ollo)ingsaying
• @hen x is zero or y and z are bothzero output is a one
• @hen x is a and y or z is a theoutput is zero
)(),,( z y x z y x f +=
),,( z y x z y x f +=
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Digital Logic :ates
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Digital Logic
• +nstead o/ true /alse or and ', )euse voltage levels3
• &ypically )e use #v and:round'3 – @e can also use #-v, #v, or others
– @e sometimes reverse the logic so that
v' and :round – &he basic principles )ont change no
matter )hat system you use3
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:round
• Eou have to have a source /orelectricity A?D a )ay /or it to return3 – &he supply line is 4o)er(!
– &he return line is :round('!
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&ransistors 0 /unction as as)itch
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And :ate
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*ultiple +nput gates (and!
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F<F and F?<&F
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&ransistor Logic
• Eou donGt have to memorize theoperations o/ the transistors and thevarious gates3
• +t%s >ust nice to have a basicunderstanding o/ ho) these thingsactually )or=3
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Boolean to Digital
• Dra) the digital circuit /rom the/ollo)ing S<4 Boolean e2uation
z xy z y x f +=),,(
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Boolean to Digital
• Dra) the digital circuit /rom the/ollo)ing S<4 Boolean e2uation
z xy z y x f +=),,(
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Boolean to Digital
• Dra) the digital circuit /rom the/ollo)ing boolean e2uation3 +s this inS<4 /orm )(),,( z y x z y x f +=
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Boolean to Digital
• Dra) the digital circuit /rom the/ollo)ing Boolean e2uation
)(),,( z y x z y x f +=
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Boolean to Digital
• Dra) the digital circuit /rom the/ollo)ing Boolean e2uation
)(),,( z y x z y x f +=
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Boolean to Digital
• Dra) the digital circuit /rom the/ollo)ing Boolean e2uation
)(),,( z y x z y x f +=
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Dra) the /ollo)ing circuits
z y z y x z y x f
z y x z y x z y x z y x z y x f
y x y x y x y x f
xy xy y x f
++=
+++=
++=
+=
)(),,(
),,(
),(
),(
C t t th t bl /
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Create a truth table /rom acircuit3
C t t th t bl /
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Create a truth table /rom acircuit3
4 4-
4 4H 4 4I
C t t th t bl /
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Create a truth table /rom acircuit3
4 A%4- B%
4 A%B4H B%#C4 (A%B!%4I (A%B!%(B%#C!
C t t th t bl /
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Create a truth table /rom acircuit3
4 A%
4- B%4 A%B4H B%#C4 (A%B!%4I (A%B!%(B%#C!
4
4
-
4 4H 4 4I
A B C A% B% A%B
B%#C
(A%B!%
(A%B!%(B%#C!
' ' '
' '
' '
'
' '
'
C t t th t bl /
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Create a truth table /rom acircuit3
4 A%
4- B%4 A%B4H B%#C4 (A%B!%4I (A%B!%(B%#C!
4
4
-
4 4H 4 4I
A B C A% B% A%B
B%#C
(A%B!%
(A%B!%(B%#C!
' ' ' ' '
' ' ' '
' ' ' ' ' '
' ' '
' ' ' '
' ' '
' ' ' ' ' '
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Jarnaugh *ap eduction
• :raphical )ay to reduce Booleane2uations
• Create truth table
• Determine size o/ Jarnaugh map
• Dra) Jarnaugh map
• 5ill in ones and zeros /rom truth table
• Circle groupings
• Determine reduced e2uation
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Size o/ *ap
• Depends on number o/ inputs – - input -x- Jarnaugh map
– input -xH Jarnaugh map
– H input HxH Jarnaugh map
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-x- Jarnaugh map
x
x
y y
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-xH Jarnaugh map
x
x
yz z y z y z y
x
x
y y
z z z
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HxH Jarnaugh map
wx
yz z y z y z y
xw
xw
xw
w
w
x
x
x
z z z
y y
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5ill in a -x- Jarnaugh map
• 1ach ro) in truth table corresponds toexactly one box in the Jarnaugh map
• 5ill in the result o/ each ro) into the
appropriate box3• &he a,b,c, and d are only included /or clarity,
you shouldnGt put them in3d c
b a
x y /(x,y!
' ' '(a!' (b!
' (c!
(d!
'
x
x
y y
x
x
y y
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5ill in a -xH Jarnaugh *ap
(h! (g! (e! (/!
(d! (c! (a! (b!
x y z /(x,y,z!
' ' ' (a!
' ' (b!
' ' (c!
' (d!
' ' (e!
' '(/!
' (g!
'(h!' '
x
x
yz z y z y z y
x
x
yz z y z y z y
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5ill in a HxH Jarnaugh map) x y z
/(x,y,
z!' ' ' ' (a!
' ' ' (b!
' ' ' (c!
' ' (d!
' ' ' (e!
' ' (/!
' ' '(g!
' '(h!
' ' ' (i!
' ' '(>!
' ' (>!
' (l!
' ' '(m!
' '(n!
' (o!
(p!
p o m n
l = i >
d c a b
h g e /
' '
'
' '
wx
yz z y z y z y
xw
xw
xw
wx
yz z y z y z y
xw
xw
xw
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Circle :roups o/ s•
All s must becircled
• 1ach circledgroup )ill be aproduct
• &he more onescircled themore reducedthe product
• Eou )ant the/e)est numbero/ big groups
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Jarnaugh eduction
' ' '
' ' '' ' ' '
' ' ' '
• All s must becircled
• 1ach circledgroup )ill be a
product• &he more ones
circled themore reducedthe product
• Eou )ant the/e)est numbero/ big groups
wx
yz z y z y z y
xw xw
xw
this?reduceyouCan
),,,( z y xw z wxy z y xw f +=
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Jarnaugh eduction
' ' '
' ' '' ' ' '
' ' ' '
• All s must becircled
• 1ach circledgroup )ill be a
product• &he more ones
circled themore reducedthe product
• Eou )ant the/e)est numbero/ big groups
wx
yz z y z y z y
xw xw
xw
z wy z y xw f =),,,(
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Jarnaugh eduction
' ' ' '
' ' ' '
' ' ' '
• All s must be circled• 1ach circled group
)ill be a product
• &he more ones
circled the morereduced the product
• Eou )ant the /e)estnumber o/ big groups
wx
yz z y z y z y
xw xw
xw
this?reduceyouCan
),,,( z y xw z y xw z y xw yz xw z y xw f +++=
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Jarnaugh eduction
' ' ' '
' ' ' '
' ' ' '
• All s must becircled
• 1ach circledgroup )ill be aproduct
• &he more onescircled the morereduced theproduct
• Eou )ant the
/e)est number o/big groups
wx
yz z y z y z y
xw xw
xw
xw z y xw f =),,,(
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Jarnaugh eduction
' '
'
' '
) x y z/(x,y,
z!' ' ' ' (a!
' ' ' (b!
' ' ' (c!
' ' (d!
' ' ' (e!
' ' (/!
' ' '(g!
' '(h!
' ' ' (i!
' ' '(>!
' ' (>!
' (l!
' ' '(m!
' '(n!
' (o!
(p!
wx
yz z y z y z y
xw
xw xw
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Jarnaugh eduction
' '
'
' '
) x y z/(x,y,
z!' ' ' ' (a!
' ' ' (b!
' ' ' (c!
' ' (d!
' ' ' (e!
' ' '(/!
' ' '(g!
' '(h!
' ' ' (i!
' ' '(>!
' ' (>!
' (l!
' ' '(m!
' '(n!
' (o!
(p!
w x
yz z y z y z y
xw
xw xw
?),,,( = z y xw f
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Jarnaugh eduction
' '
'
' '
) x y z/(x,y,
z!' ' ' ' (a!
' ' ' (b!
' ' ' (c!
' ' (d!
' ' ' (e!
' ' '(/!
' ' '(g!
' '(h!
' ' ' (i!
' ' '(>!
' ' (>!
' (l!
' ' '(m!
' '(n!
' (o!
(p!
w x
yz z y z y z y
xw
xw xw
yw xw z xwy z y xw f +++=),,,(
Eou can go oK the edges
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Eou can go oK the edges
' ' ' '
' ' ' '
' '
' '
' ' ' '
' ' ' '
' '
' ' ' '
' ' ' '
' '
wx
yz z y z y z y
xw
xw
xw
wx
yz z y z y z y
xw
xw
xw
wx
yz z y z y z y
xw
xw
xw
z w z y xw f =),,,( x z y xw f =),,,(
xz z y xw f =),,,(
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5ind the reduced e2uation
x y /(x,y!
' ' '
'
'
x y z /(x,y,z!
' ' '
' '
' '
'
' '
' '
' '
'
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Car @arning
• @arning beep prevents – Jeys le/t in ignition
– Lights le/t on
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Car @arning
• &hree inputs – Lights(l!
• lights on
•
lights oK' – Jey(=!
• =ey in
• =ey out'
– door(d!• door open
• door closed'
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Create a truth table
l = d b' ' '
' '
' '
'
' '
'
'
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Create a truth table
l = d b' ' ' '
' ' '
' ' '
'
' '
'
' '
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Create a Jarnaugh map
l(x! =(y! d(z! b
' ' ' '
' ' '
' ' '
'
' '
'
' '
(h! (g! (e! (/!
(d! (c! (a! (b!
x
x
yz z y z y z y
x
x
yz z y z y z y
7/23/2019 03 Boolean Algebra Digital Logic
http://slidepdf.com/reader/full/03-boolean-algebra-digital-logic 73/79
Create a Jarnaugh map
l(x! =(y! d(z! b
' ' ' '
' ' '
' ' '
'
' '
'
' '
(h! (g! (e! (/!
(d! (c! (a! (b!
'
' ' '
x
x
yz z y z y z y
x
x
yz z y z y z y
7/23/2019 03 Boolean Algebra Digital Logic
http://slidepdf.com/reader/full/03-boolean-algebra-digital-logic 74/79
Circle :roups
l(x! =(y! d(z! b
' ' ' '
' ' '
' ' '
'
' '
'
' '
(h! (g! (e! (/!
(d! (c! (a! (b!
'
' ' '
x
x
yz z y z y z y
x
x
yz z y z y z y
7/23/2019 03 Boolean Algebra Digital Logic
http://slidepdf.com/reader/full/03-boolean-algebra-digital-logic 75/79
Circle :roups
l(x! =(y! d(z! b
' ' ' '
' ' '
' ' '
'
' '
'
' '
(h! (g! (e! (/!
(d! (c! (a! (b!
'
' ' '
x
x
yz z y z y z y
x
x
yz z y z y z y
Determine the Boolean
7/23/2019 03 Boolean Algebra Digital Logic
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Determine the Boolean/unction
l(x! =(y!
d(z!
/(l,=,d!
' ' ' '
' ' '' ' '
'
' '
'
' '
'
' ' '
l
l
kd d k d k d k
?),,( =d k l f
Determine the Boolean
7/23/2019 03 Boolean Algebra Digital Logic
http://slidepdf.com/reader/full/03-boolean-algebra-digital-logic 77/79
Determine the Boolean/unction
l(x! =(y!
d(z!
/(l,=,d!
' ' ' '
' ' '' ' '
'
' '
'
' '
'
' ' '
l
l
kd d k d k d k
k l kd d k l f +=),,(
h i i
7/23/2019 03 Boolean Algebra Digital Logic
http://slidepdf.com/reader/full/03-boolean-algebra-digital-logic 78/79
Dra) the circuit
k l kd d k l f +=),,(
i Ci i
7/23/2019 03 Boolean Algebra Digital Logic
http://slidepdf.com/reader/full/03-boolean-algebra-digital-logic 79/79
Design a Circuit
• A digital circuit has a /our bitnumerical input ('.!
• Design a circuit )hich )ill detect the
numbers . inclusive