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Young Won Lim9/18/13
Truth Table (2A)
Young Won Lim9/18/13
Copyright (c) 2011-2013 Young W. Lim.
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Truth Table (2A) 3 Young Won Lim9/18/13
Truth Table
x
y
z?
1
0
1
0
0
0
0
1
0 1 1
0 1 0
0 0 1
0 0 0
1 1 1
1 1 0
1 0 1
1 0 0
inputs output
x y z F
F
I/O relationship
Truth Table (2A) 4 Young Won Lim9/18/13
Truth Table and minterms (1)
0 1 1
0 1 0
0 0 1
0 0 0
1 1 1
1 1 0
1 0 1
1 0 0
the case when x=0 and y=1 and z=1
the case when x=0 and y=1 and z=0
the case when x=0 and y=0 and z=0
the case when x=0 and y=0 and z=1
the case when x=1 and y=1 and z=1
the case when x=1 and y=1 and z=0
the case when x=1 and y=0 and z=0
the case when x=1 and y=0 and z=1
inputs
All possible combination of inputs
x y z
x y z = 1
x y z = 1
x y z = 1
x y z = 1
x y z = 1
x y z = 1
x y z = 1
x y z = 1
x y z = 1
For the output of an and gate to be 1, all inputs must be 1
x=1y=1
z=1
x=1y=0
z=11
Truth Table (2A) 5 Young Won Lim9/18/13
Truth Table and minterms (2)
0 1 1
0 1 0
0 0 1
0 0 0
1 1 1
1 1 0
1 0 1
1 0 0
the case when the minterm
the case when the minterm
the case when the minterm
the case when the minterm
the case when the minterm
the case when the minterm
the case when the minterm
the case when the minterm
inputs
All possible combination of inputs
x y z
x y z = 1
x y z = 1
x y z = 1
x y z = 1
x y z = 1
x y z = 1
x y z = 1
x y z = 1
3
2
0
1
7
6
4
5
index
m0 =
m1 =
m2 =
m3 =
m4 =
m5 =
m6 =
m7 =
m5minterm index 5
binary 101
x y z = 1
0
bar
x=1y=1
z=1
1
Truth Table (2A) 6 Young Won Lim9/18/13
Truth Table and MAXterms (1)
0 1 1
0 1 0
0 0 1
0 0 0
1 1 1
1 1 0
1 0 1
1 0 0
the case when x=0 and y=1 and z=1
the case when x=0 and y=1 and z=0
the case when x=0 and y=0 and z=0
the case when x=0 and y=0 and z=1
the case when x=1 and y=1 and z=1
the case when x=1 and y=1 and z=0
the case when x=1 and y=0 and z=0
the case when x=1 and y=0 and z=1
inputs
All possible combination of inputs
x y z
x+ y+z = 0
x+ y+ z = 0
x+ y+z = 0
x+ y+ z = 0
x+ y+z = 0
x+ y+ z = 0
x+ y+z = 0
x+ y+ z = 0
For the output of an or gate to be 0, all inputs must be 0
x=0y=0
z=0
x+ y+ z = 0
x=1y=0
z=10
Truth Table (2A) 7 Young Won Lim9/18/13
Truth Table and MAXterms (2)
0 1 1
0 1 0
0 0 1
0 0 0
1 1 1
1 1 0
1 0 1
1 0 0
the case when the MAXterm
the case when the MAXterm
the case when the MAXterm
the case when the MAXterm
the case when the MAXterm
the case when the MAXterm
the case when the MAXterm
the case when the MAXterm
inputs
All possible combination of inputs
x y z
3
2
0
1
7
6
4
5
index
M 0 =
M 1 =
M 2 =
M 3 =
M 4 =
M 5 =
M 6 =
M 7 =
M 5minterm index 5
binary 101 1
bar
x+ y+z = 0
x+ y+ z = 0
x+ y+z = 0
x+ y+ z = 0
x+ y+z = 0
x+ y+ z = 0
x+ y+z = 0
x+ y+ z = 0
x+ y+ z
x=0y=0
z=00
Truth Table (2A) 8 Young Won Lim9/18/13
1
0
0
0
xy
Maxterm and minterm Conditions
1 1
1 0
0 1
0 0
x y
1
1
1
0
x+y
1 1
1 0
0 1
0 0
x y
0
0
0
1
xy
0 0
0 1
1 0
1 1
x y
0
1
1
1
x+y
0 0
0 1
1 0
1 1
x y
1
01
01
1
1
1
0
0
0
0
Truth Table (2A) 9 Young Won Lim9/18/13
1
0
1
0
0
0
0
1
Boolean Function with minterms (1)
0 1 1
0 1 0
0 0 1
0 0 0
1 1 1
1 1 0
1 0 1
1 0 0
inputs output
All possible combination of inputs
x y z
3
2
0
1
7
6
4
5
index
F The output F becomes 1, for one of the three following cases
(the case when x=0 and y=1 and z=1)
(the case when x=0 and y=0 and z=1)
(the case when x=1 and y=0 and z=0)
or
or
x y z = 1
x y z = 1
x y z = 1
m1 =
m3 =
m4 =
Domain Range
m1
m3
m4
m0
m2
m5
m6
m7
1
0
Truth Table (2A) 10 Young Won Lim9/18/13
1
0
1
0
0
0
0
1
Boolean Function with minterms (2)
0 1 1
0 1 0
0 0 1
0 0 0
1 1 1
1 1 0
1 0 1
1 0 0
inputs output
All possible combination of inputs
x y z
3
2
0
1
7
6
4
5
index
F
F = m1 +m3 +m4
The output F becomes 1, m1=1either m3=1 m4=1or or
For the output of an or gate to be 1, at least one must be 1
m1 +m3 +m4=1 F = 1
m1
m3
m4
Fm0m2
m7
Fm5m6
Truth Table (2A) 11 Young Won Lim9/18/13
1
0
1
0
0
0
0
1
Boolean Function with minterms (3)
0 1 1
0 1 0
0 0 1
0 0 0
1 1 1
1 1 0
1 0 1
1 0 0
inputs output
All possible combination of inputs
x y z
3
2
0
1
7
6
4
5
index
F
F = m1 +m3 +m4
The output F becomes 1, m1=1either m3=1 m4=1or or
For the output of an or gate to be 1, at least one must be 1
m1 +m3 +m4=1 F = 1
The output F becomes 0, either
F = 0
m0=1 m2=1 m5=1 m6=1 m7=1or or or or
m0+m2+m5+m6+m7=1
F = m0+m2+m5+m6+m7
Truth Table (2A) 12 Young Won Lim9/18/13
(the case when x=0 and y=1 and z=0)
(the case when x=0 and y=0 and z=0)
(the case when x=1 and y=1 and z=1)
(the case when x=1 and y=1 and z=0)
(the case when x=1 and y=0 and z=1)
1
0
1
0
0
0
0
1
Boolean Function with Maxterms (1)
0 1 1
0 1 0
0 0 1
0 0 0
1 1 1
1 1 0
1 0 1
1 0 0
inputs output
All possible combination of inputs
x y z
3
2
0
1
7
6
4
5
index
FThe output F becomes 0, for one of the five following cases
or
or
Domain
RangeM 1
M 3
M 4
M 0
M 2
M 5
M 6
M 7
1
0
x+ y+z = 0
x+ y+z = 0
x+ y+ z = 0
x+ y+z = 0
x+ y+ z = 0
or
or
ororor
Truth Table (2A) 13 Young Won Lim9/18/13
1
0
1
0
0
0
0
1
Boolean Function with Maxterms (2)
0 1 1
0 1 0
0 0 1
0 0 0
1 1 1
1 1 0
1 0 1
1 0 0
inputs output
All possible combination of inputs
x y z
3
2
0
1
7
6
4
5
index
FThe output F becomes 0, M 0=0either M 2=0 M 5=0 M 6=0 M 7=0or or or or
For the output of an and gate to be 0, at least one input must be 0
F = 0M 0⋅M 2⋅M 5⋅M 6⋅M 7=0
F = M 0⋅M 2⋅M 5⋅M 6⋅M 7
M 0M 2
M 7
M 5M 6
M 1
M 3
M 4
F F
Truth Table (2A) 14 Young Won Lim9/18/13
1
0
1
0
0
0
0
1
Boolean Function with Maxterms (2)
0 1 1
0 1 0
0 0 1
0 0 0
1 1 1
1 1 0
1 0 1
1 0 0
inputs output
All possible combination of inputs
x y z
3
2
0
1
7
6
4
5
index
FThe output F becomes 0, M 0=0either M 2=0 M 5=0 M 6=0 M 7=0or or or or
For the output of an and gate to be 0, at least one input must be 0
F = 0
The output F becomes 1, either
F = 1
M 0⋅M 2⋅M 5⋅M 6⋅M 7=0
F = M 0⋅M 2⋅M 5⋅M 6⋅M 7
M 1=0 M 3=0 M 4=0or or
M 1⋅M 3⋅M 4=0
F = M 1⋅M 3⋅M 4
Truth Table (2A) 15 Young Won Lim9/18/13
1
0
1
0
0
0
0
1
Complimentary Relations
0 1 1
0 1 0
0 0 1
0 0 0
1 1 1
1 1 0
1 0 1
1 0 0
inputs output
All possible combination of inputs
x y z
3
2
0
1
7
6
4
5
index
F
The output F becomes 0,
F (x , y , z) = M 0⋅M 2⋅M 5⋅M 6⋅M 7
M 0=0either M 2=0 M 5=0 M 6=0 M 7=0or or or or
For the output of an and gate to be 0, at least one input must be 0
F (x , y , z) = m1 + m3 + m4
m1=1either m3=1 m4=1or or
For the output of an or gate to be 1, at least one must be 1
The output F becomes 1,
F (x , y , z) = m0 + m2 + m5 + m6 + m7
F (x , y , z) = m0 + m2 + m5 + m6 + m7
= m0⋅m2⋅m5⋅m6⋅m7
mi = M i
M i = mi
Truth Table (2A) 16 Young Won Lim9/18/13
Boolean Function Summary
for the cases
The output F becomes 1,
1) when
M 1=02) when M 3=0 M 4=0or or
F=1
m1=1 m3=1 m4=1or or
F (x , y , z) = m1 + m3 + m4
F=1F (x , y , z) = M 1⋅M 3⋅M 4 (F=0)
for the cases
The output F becomes 0,
1) when
2) when
F=0
F=0
(F=1)
1
1
10 1 10 1 00 0 10 0 0
1 1 11 1 01 0 11 0 0
x y z
3
2
0
1
7
6
4
5
F
0
0
000
0 1 10 1 00 0 10 0 0
1 1 11 1 01 0 11 0 0
x y z
3
2
0
1
7
6
4
5
F
M 0=0 M 2=0 M 5=0 M 6=0 M 7=0or or or or
F (x , y , z) = M 0⋅M 2⋅M 5⋅M 6⋅M 7
m0=1 m2=1 m5=1 m6=1 m7=1or or or or
F (x , y , z) = m0+m2+m5+m6+m7
Truth Table (2A) 17 Young Won Lim9/18/13
Boolean Function Summary
F=1F (x , y , z) = m1 + m3 + m4
F=0
F=0
(F=1)
0 1 10 1 00 0 10 0 0
1 1 11 1 01 0 11 0 0
x y z
3
2
0
1
7
6
4
5
F
0
0
000
0 1 10 1 00 0 10 0 0
1 1 11 1 01 0 11 0 0
x y z
3
2
0
1
7
6
4
5
F
F (x , y , z) = M 0⋅M 2⋅M 5⋅M 6⋅M 7
F (x , y , z) = m0+m2+m5+m6+m7
1
1
1
0
0
000
1
1
1
F=1F (x , y , z) = M 1⋅M 3⋅M 4 (F=0)
Truth Table (2A) 18 Young Won Lim9/18/13
SOP and POS
0 1 10 1 00 0 10 0 0
1 1 11 1 01 0 11 0 0
AND
AND
AND
AND
0 1 10 1 00 0 10 0 0
1 1 11 1 01 0 11 0 0
OR
0 1 10 1 00 0 10 0 0
1 1 11 1 01 0 11 0 0
AND
AND
AND
AND
0 1 10 1 00 0 10 0 0
1 1 11 1 01 0 11 0 0
OR
1
0
1
0
SOP
POS
Truth Table (2A) 19 Young Won Lim9/18/13
1
0
1
0
0
0
0
1
Boolean Function with minterms
0 1 1
0 1 0
0 0 1
0 0 0
1 1 1
1 1 0
1 0 1
1 0 0
x y z
3
2
0
1
7
6
4
5
F
1
11
11
0
1
0
0
0
0
1
0 1 1
0 1 0
0 0 1
0 0 0
1 1 1
1 1 0
1 0 1
1 0 0
x y z
3
2
0
1
7
6
4
5
F1
11
1
1
1
F F
Truth Table (2A) 20 Young Won Lim9/18/13
1
0
1
0
0
0
0
1
Boolean Function with Maxterms
0 1 1
0 1 0
0 0 1
0 0 0
1 1 1
1 1 0
1 0 1
1 0 0
x y z
3
2
0
1
7
6
4
5
F
0
00
01
0
1
0
0
0
0
1
0 1 1
0 1 0
0 0 1
0 0 0
1 1 1
1 1 0
1 0 1
1 0 0
x y z
3
2
0
1
7
6
4
5
F0
0
0
0
0
FF
0
Truth Table (2A) 21 Young Won Lim9/18/13
Truth Table
Young Won Lim9/18/13
References
[1] http://en.wikipedia.org/
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