remember and be thankful
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ECEN 301 Discussion #22 – Combinational Logic 1
Remember and be Thankful2 Nephi 1:9, 20: 9 Wherefore, I, Lehi, have obtained a promise, that inasmuch as those whom
the Lord God shall bring out of the land of Jerusalem shall keep his commandments, they shall prosper upon the face of this land; and they shall be kept from all other nations, that they may possess this land unto themselves. And if it so be that they shall keep his commandments they shall be blessed upon the face of this land, and there shall be none to molest them, nor to take away the land of their inheritance; and they shall dwell safely forever.
20 And he hath said that: Inasmuch as ye shall keep my commandments ye shall prosper in the land; but inasmuch as ye will not keep my commandments ye shall be cut off from my presence.
ECEN 301 Discussion #22 – Combinational Logic 2
Lecture 22 – Boolean Algebra & Combinational Logic
ECEN 301 Discussion #22 – Combinational Logic 3
Boolean AlgebraBoolean Algebra: the mathematics associated with
binary numbers• Developed by George Boole in 1854
Variables in boolean algebra can take only one of two possible values:0 → FALSE1 → TRUE
ECEN 301 Discussion #22 – Combinational Logic 4
Rules of Boolean Algebra
ZXYXZXZYYXYXYXX
ZYXZXYXXYXXXZXX
.19
.18)()(.17
)(.61.15
XX
XXXXXXX
XXX
XXXX
XX
.9
0.8.7
1.600.5
1.4.3
11.20.1
ZXYXZYXZYXZYX
ZYXZYXXYYXXYYX
)(.14)()(.13
)()(.12.11.10
ECEN 301 Discussion #22 – Combinational Logic 5
DeMorgan’s Law
BABA
BABA
To distribute the bar,change the operation.
NOR Symbols
NAND Symbols
ECEN 301 Discussion #22 – Combinational Logic 6
Boolean AlgebraExample1: simplify the following function
ACDBCDBDADBAOUT
ECEN 301 Discussion #22 – Combinational Logic 7
Boolean AlgebraExample1: simplify the following function
CADOUTDACDOUT
DABCDOUTBCDCDDAOUTBCDCADOUTBCDACADOUTACDBCDDAOUT
ACDBCDBBDAOUTACDBCDBDADBAOUT
14 Rule2 Rule)1(14 Rule14 Rule18 Rule14 Rule4 Rule14 Rule
ECEN 301 Discussion #22 – Combinational Logic 8
Boolean AlgebraExample2: Simplify the equation created by the
following truth table
A B C Z0 0 0 00 0 1 10 1 0 00 1 1 11 0 0 11 0 1 11 1 0 11 1 1 1
ECEN 301 Discussion #22 – Combinational Logic 9
Boolean AlgebraExample2: Simplify the equation created by the
following truth table
A B C Z0 0 0 00 0 1 10 1 0 00 1 1 11 0 0 11 0 1 11 1 0 11 1 1 1
ABCCABCBACBABCACBAZ
ECEN 301 Discussion #22 – Combinational Logic 10
Boolean AlgebraExample2: Simplify the equation created by the
following truth table
A B C Z0 0 0 00 0 1 10 1 0 00 1 1 11 0 0 11 0 1 11 1 0 11 1 1 1
CAZACAZ
BBACAZ
ABBACAZ
CCABCCBABBCAZ
ABCCABCBACBABCACBAZ
ECEN 301 Discussion #22 – Combinational Logic 11
Boolean AlgebraExample3: Determine the truth table
A B C Z0 0 0 ?0 0 1 ?0 1 0 ?0 1 1 ?1 0 0 ?1 0 1 ?1 1 0 ?1 1 1 ?
A
B
C
Z
ECEN 301 Discussion #22 – Combinational Logic 12
Boolean AlgebraExample3: Determine the truth table
A B C x1 Z0 0 0 0 ?0 0 1 0 ?0 1 0 1 ?0 1 1 1 ?1 0 0 1 ?1 0 1 1 ?1 1 0 1 ?1 1 1 1 ?
A
B
C
Z
BAx 1
ECEN 301 Discussion #22 – Combinational Logic 13
Boolean AlgebraExample3: Determine the truth table
A B C x1 x2 Z0 0 0 0 1 ?0 0 1 0 1 ?0 1 0 1 1 ?0 1 1 1 0 ?1 0 0 1 1 ?1 0 1 1 1 ?1 1 0 1 1 ?1 1 1 1 0 ?
A
B
C
Z
BAx 1
BCx 2
ECEN 301 Discussion #22 – Combinational Logic 14
Boolean AlgebraExample3: Determine the truth table
A B C x1 x2 x3 Z0 0 0 0 1 0 ?0 0 1 0 1 1 ?0 1 0 1 1 0 ?0 1 1 1 0 1 ?1 0 0 1 1 1 ?1 0 1 1 1 1 ?1 1 0 1 1 1 ?1 1 1 1 0 1 ?
A
B
C
Z
BAx 1
BCx 2CAx 3
ECEN 301 Discussion #22 – Combinational Logic 15
Boolean AlgebraExample3: Determine the truth table
A B C x1 x2 x3 Z0 0 0 0 1 0 00 0 1 0 1 1 00 1 0 1 1 0 00 1 1 1 0 1 01 0 0 1 1 1 11 0 1 1 1 1 11 1 0 1 1 1 11 1 1 1 0 1 0
A
B
C
Z
BAx 1
BCx 2CAx 3 CABCBAZ
ECEN 301 Discussion #22 – Combinational Logic 16
Boolean AlgebraExample3: Determine the truth table
A B C x1 x2 x3 Z0 0 0 0 1 0 00 0 1 0 1 1 00 1 0 1 1 0 00 1 1 1 0 1 01 0 0 1 1 1 11 0 1 1 1 1 11 1 0 1 1 1 11 1 1 1 0 1 0
BCA
CBCA
CBACA
CBABBCA
CABCBACBA
CABCBAZ
ECEN 301 Discussion #22 – Combinational Logic 17
Combinational Logic
DecodersMultiplexers
ECEN 301 Discussion #22 – Combinational Logic 18
Decoders
• with n inputs has 2n outputs
X
Y
Z
W
2-to-4Decoder
A B
W
X
Y
Z
DECODERSymbol
ECEN 301 Discussion #22 – Combinational Logic 19
Decoders• Write the truth table
X
Y
Z
W
ECEN 301 Discussion #22 – Combinational Logic 20
Decoders• Write the truth table
X
Y
Z
W
A B W X Y Z0 0 1 0 0 0
0 1 0 1 0 0
1 0 0 0 1 0
1 1 0 0 0 1
ECEN 301 Discussion #22 – Combinational Logic 21
Multiplexors• Connect one of its inputs to its output according to
select signals
• Useful for selecting one from a collection of data inputs.
• Usually has 2n inputs and n select lines.
A B
S
C
1 0
MULTIPLEXOR Symbol
ECEN 301 Discussion #22 – Combinational Logic 22
Multiplexors• Write the truth table
A B
S
C
1 0
MULTIPLEXOR Symbol
A B S C0 0 0 ?
0 0 1 ?
0 1 0 ?
0 1 1 ?
1 0 0 ?
1 0 1 ?
1 1 0 ?
1 1 1 ?
ECEN 301 Discussion #22 – Combinational Logic 23
Multiplexors• Write the truth table
A B
S
C
1 0
MULTIPLEXOR Symbol
A B S C0 0 0 0
0 0 1 0
0 1 0 1
0 1 1 0
1 0 0 0
1 0 1 1
1 1 0 1
1 1 1 1
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