Young Won Lim3/19/18
Copyright (c) 2015 – 2018 Young W. Lim.
Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License".
Please send corrections (or suggestions) to [email protected].
This document was produced by using LibreOffice and Octave.
Logic (7A)Resolution 16 Young Won Lim
3/19/18
Example B – (1)
p → q∨r
p∨¬q
r∨q
¬p∨q∨r
p∨¬q
r∨q
Discrete Mathematics, Johnsonbough
q∨¬q∨r
r∨q
r∨q
q∨¬q∨r = T∨r = T
p → q∨r
p∨¬q
r∨q
¬p∨q∨r
p∨¬q
r∨q
p∨r
Logic (7A)Resolution 17 Young Won Lim
3/19/18
Example B – (2)
p → q∨r
p∨¬q
r∨q
p → q∨r
¬p → ¬q
q∨r
Discrete Mathematics, Johnsonbough
p → q∨r
q → p
q∨r
p∨r
Logic (7A)Resolution 18 Young Won Lim
3/19/18
p q r ¬p ¬p∨q∨r
T T T F T
T T F F T
T F T F T
T F F F F
F T T T T
F T F T T
F F T T T
F F F T T
p q r ¬q p∨¬q
T T T F T
T T F F T
T F T T T
T F F T T
F T T F F
F T F F F
F F T T T
F F F T T
Truth Table
p q r q∨r
T T T T
T T F T
T F T T
T F F F
F T T T
F T F T
F F T T
F F F F
p q r ¬p∨q∨r p∨¬q q∨r H 1∧H 2∧H 3 p∨rT T T T T T T T
T T F T T T T T
T F T T T T T T
T F F F T F F TF T T T F T F T
F T F T F T F F
F F T T T T T TF F F T T F F F
H 1 = ¬p∨q∨r
H 2 = p∨¬q
H 3 = q∨r
H 1∧H 2∧H 3 → H 3
H 1∧H 2∧H 3 → H 2
H 1∧H 2∧H 3 → H 1
H 1∧H 2∧H 3 → (p∨r )
Logic (7A)Resolution 19 Young Won Lim
3/19/18
p q r H 1∨H 2∨H 3
T T T T
T T F T
T F T T
T F F F
F T T F
F T F F
F F T T
F F F F
Truth Table and K-Map
p q r H 1∨H 2∨H 3
1 1 1 1
1 1 0 1
1 0 1 1
1 0 0 0
0 1 1 0
0 1 0 0
0 0 1 1
0 0 0 0
p q r H 1∨H 2∨H 3
0 0 0 0
0 0 1 1
0 1 0 0
0 1 1 0
1 0 0 0
1 0 1 1
1 1 0 1
1 1 1 1
0 0 0 1 1 1 1 0
0 1 0 0
0 1 1 1
0
1
p q r
Logic (7A)Resolution 20 Young Won Lim
3/19/18
K-Map and Logic Minimization
0 0 0 1 1 1 1 0
0 1 0 0
0 1 1 1
0
1
p q r
0 0 0 1 1 1 1 0
0
1
p q r
p q r pq r pq r
0 0 0 1 1 1 1 0
0
1q r
p q
p q r + p q r = ( p + p)q r = q r
p q r + pq r = pq(r + r) = p q
Logic (7A)Resolution 21 Young Won Lim
3/19/18
K-Map : Verification
0 0 0 1 1 1 1 0
0
1q r
p qH 1∧H 2∧H 3 ≡ q r + pq
p q r q q r pq q r+ pq
0 0 0 1 0 0 0
0 0 1 1 1 0 1
0 1 0 0 0 0 0
0 1 1 0 0 0 0
1 0 0 1 0 0 0
1 0 1 1 1 0 1
1 1 0 0 0 1 1
1 1 1 0 0 1 1
p q r H 1∨H 2∨H 3
0 0 0 0
0 0 1 1
0 1 0 0
0 1 1 0
1 0 0 0
1 0 1 1
1 1 0 1
1 1 1 1
Logic (7A)Resolution 22 Young Won Lim
3/19/18
Adding two don’t care conditions
0 0 0 1 1 1 1 0
0 1 X X
0 1 1 1
0
1
p q r
0 0 0 1 1 1 1 0
0
1r
0 0 0 1 1 1 1 0
0
1p
0 0 0 1 1 1 1 0
0 1 X X
0 1 1 1
0
1
p q r
p∨r
Logic (7A)Resolution 23 Young Won Lim
3/19/18
K-Map : Verification
H 1∧H 2∧H 3 → p∨r
p q r p∨r
0 0 0 0
0 0 1 1
0 1 0 0
0 1 1 1
1 0 0 1
1 0 1 1
1 1 0 1
1 1 1 1
p q r H 1∨H 2∨H 3 p∨q
0 0 0 0 0
0 0 1 1 1
0 1 0 0 0
0 1 1 0 1
1 0 0 0 1
1 0 1 1 1
1 1 0 1 1
1 1 1 1 1
0 0 0 1 1 1 1 0
0
1r
0 0 0 1 1 1 1 0
0
1p
Young Won Lim3/22/18
Copyright (c) 2015 – 2018 Young W. Lim.
Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License".
Please send corrections (or suggestions) to [email protected].
This document was produced by using LibreOffice and Octave.
Logic (8A)Boolean Algebra
4 Young Won Lim3/22/18
Boolean Algebra
https://en.wikipedia.org/wiki/Boolean_algebra
Logic (8A)Boolean Algebra
5 Young Won Lim3/22/18
Operators
https://en.wikipedia.org/wiki/Boolean_algebra
xy x+y x
Logic (8A)Boolean Algebra
6 Young Won Lim3/22/18
Laws (1)
https://en.wikipedia.org/wiki/Boolean_algebra
x+(y+z) = (x+y)+z
x(yz) = (xy)zx+y = y+x
xy = yz
x(y+z) = xy + xz
x+0=x
x*1=x
x*0=0
Logic (8A)Boolean Algebra
7 Young Won Lim3/22/18
Laws (2)
https://en.wikipedia.org/wiki/Boolean_algebra
x+1=1
x+x=xx*x=x
x(x+y)=x
x+xy=x
x+yz=(x+y)(x+z)
xx=0
x+x=1
x y = (x+y)
(x+y) = xy
Logic (8A)Boolean Algebra
8 Young Won Lim3/22/18
Digital Logic Gates
https://en.wikipedia.org/wiki/Boolean_algebra
Logic (8A)Boolean Algebra
10 Young Won Lim3/22/18
AND, OR Gates
https://en.wikipedia.org/wiki/Logic_gate
Logic (8A)Boolean Algebra
11 Young Won Lim3/22/18
NAND, NOR Gates
https://en.wikipedia.org/wiki/Logic_gate
Logic (8A)Boolean Algebra
12 Young Won Lim3/22/18
XOR, XNOR Gates
https://en.wikipedia.org/wiki/Logic_gate
Logic (8A)Boolean Algebra
13 Young Won Lim3/22/18
CMOS Logic Gates
https://en.wikipedia.org/wiki/CMOS
Logic (8A)Boolean Algebra
14 Young Won Lim3/22/18
Identity and Null Element Theorem
0 1
0 1
0 1
x
x
x
x
x
x
x
x
x
x
0
1
x⋅0 = 0 x⋅1 = x
x+ 0 = x x+ 1 = 1
x+ 0 = x x+ 1 = x
https://en.wikiversity.org/wiki/Discrete_Mathematics_in_plain_view#Algorithms
Logic (8A)Boolean Algebra
15 Young Won Lim3/22/18
Distributive
x⋅( y + z) = x⋅ y + x⋅z
x + ( y⋅ z) = (x + y)⋅(x + z)
≠ x⋅ y + z
= x + y⋅ z
This parenthesis cannot be deleted
This parenthesis can be deleted
Operator precedence : ⋅ > +
https://en.wikiversity.org/wiki/Discrete_Mathematics_in_plain_view#Algorithms
Logic (8A)Boolean Algebra
16 Young Won Lim3/22/18
Inclusion
x⋅(x + y ) = x
= x⋅(1 + y)
= x
x y
x + y
x + xy = x
x + xy = x⋅1 + x⋅ y
= x⋅(1 + y )
= x
x y
x y
x⋅(x + y ) = x⋅ x + x⋅ y
= x + x⋅ y
https://en.wikiversity.org/wiki/Discrete_Mathematics_in_plain_view#Algorithms
Logic (8A)Boolean Algebra
17 Young Won Lim3/22/18
Inclusion
x⋅(x + y ) = x
= x⋅(1 + y)
= x
x y
x + y
x + xy = x
x + xy = x⋅1 + x⋅ y
= x⋅(1 + y )
= x
x y
x y
x⋅(x + y ) = x⋅ x + x⋅ y
= x + x⋅ y
https://en.wikiversity.org/wiki/Discrete_Mathematics_in_plain_view#Algorithms
Logic (8A)Boolean Algebra
18 Young Won Lim3/22/18
`
Eliminate
x⋅(x + y ) = x y
= x⋅ y
x y
x + y
x + x y = x + y
= 1⋅(x + y )
= x + y
x y
x y
x⋅(x + y ) = x⋅ x + x⋅ y
= 0 + x⋅ y
x + x y = (x + x )⋅(x + y)
https://en.wikiversity.org/wiki/Discrete_Mathematics_in_plain_view#Algorithms
Young Won Lim3/22/18
Copyright (c) 2015 – 2018 Young W. Lim.
Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License".
Please send corrections (or suggestions) to [email protected].
This document was produced by using LibreOffice and Octave.
Logic (8B)Boolean Functions
3 Young Won Lim3/22/18
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
https://en.wikiversity.org/wiki/The_necessities_in_Digital_Design
Logic (8B)Boolean Functions
4 Young Won Lim3/22/18
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=1
y=1
z=1
x=1
y=0
z=1
1
https://en.wikiversity.org/wiki/The_necessities_in_Digital_Design
Logic (8B)Boolean Functions
5 Young Won Lim3/22/18
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=1
y=1
z=1
1
https://en.wikiversity.org/wiki/The_necessities_in_Digital_Design
Logic (8B)Boolean Functions
6 Young Won Lim3/22/18
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=0
y=0
z=0
x+ y+ z = 0
x=1
y=0
z=1
0
https://en.wikiversity.org/wiki/The_necessities_in_Digital_Design
Logic (8B)Boolean Functions
7 Young Won Lim3/22/18
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
M0
=
M1
=
M2
=
M3
=
M4
=
M5
=
M6
=
M7
=
M5minterm 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=0
y=0
z=0
0
https://en.wikiversity.org/wiki/The_necessities_in_Digital_Design
Logic (8B)Boolean Functions
8 Young Won Lim3/22/18
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
https://en.wikiversity.org/wiki/The_necessities_in_Digital_Design
Logic (8B)Boolean Functions
9 Young Won Lim3/22/18
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
https://en.wikiversity.org/wiki/The_necessities_in_Digital_Design
Logic (8B)Boolean Functions
10 Young Won Lim3/22/18
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+ m
3+ m
4
The output F becomes 1, m
1=1either m
3=1 m
4=1or or
For the output of an or gate to be 1, at least one must be 1
m1+ m
3+ m
4=1 F = 1
m1
m3
m4
Fm
0
m2
m7
Fm
5
m6
https://en.wikiversity.org/wiki/The_necessities_in_Digital_Design
Logic (8B)Boolean Functions
11 Young Won Lim3/22/18
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+ m
3+ m
4
The output F becomes 1, m
1=1either m
3=1 m
4=1or or
For the output of an or gate to be 1, at least one must be 1
m1+ m
3+ m
4=1 F = 1
The output F becomes 0, either
F = 0
m0=1 m
2=1 m
5=1 m
6=1 m
7=1or or or or
m0+m
2+m
5+m
6+m
7=1
F = m0+m
2+m
5+m
6+m
7
https://en.wikiversity.org/wiki/The_necessities_in_Digital_Design
Logic (8B)Boolean Functions
12 Young Won Lim3/22/18
(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
F
The output F becomes 0, for one of the five following cases
or
or
Domain
Range
M1
M3
M4
M0
M2
M5
M6
M7
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
https://en.wikiversity.org/wiki/The_necessities_in_Digital_Design
Logic (8B)Boolean Functions
13 Young Won Lim3/22/18
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 = 0M0⋅M
2⋅M
5⋅M
6⋅M
7=0
F = M0⋅M
2⋅M
5⋅M
6⋅M
7
M0
M2
M7
M5
M6
M1
M3
M4
F F
https://en.wikiversity.org/wiki/The_necessities_in_Digital_Design
Logic (8B)Boolean Functions
14 Young Won Lim3/22/18
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
M0⋅M
2⋅M
5⋅M
6⋅M
7=0
F = M0⋅M
2⋅M
5⋅M
6⋅M
7
M1=0 M
3=0 M
4=0or or
M1⋅M
3⋅M
4=0
F = M1⋅M
3⋅M
4
https://en.wikiversity.org/wiki/The_necessities_in_Digital_Design
Logic (8B)Boolean Functions
15 Young Won Lim3/22/18
SOP and POS
0 1 1
0 1 0
0 0 1
0 0 0
1 1 1
1 1 0
1 0 1
1 0 0
AND
AND
AND
AND
0 1 1
0 1 0
0 0 1
0 0 0
1 1 1
1 1 0
1 0 1
1 0 0
OR
0 1 1
0 1 0
0 0 1
0 0 0
1 1 1
1 1 0
1 0 1
1 0 0
AND
AND
AND
AND
0 1 1
0 1 0
0 0 1
0 0 0
1 1 1
1 1 0
1 0 1
1 0 0
OR
1
0
1
0
SOP
POS
https://en.wikiversity.org/wiki/The_necessities_in_Digital_Design
sum+
product*
sum+
product*
Logic (8B)Boolean Functions
16 Young Won Lim3/22/18
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
https://en.wikiversity.org/wiki/The_necessities_in_Digital_Design
Logic (8B)Boolean Functions
17 Young Won Lim3/22/18
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
https://en.wikiversity.org/wiki/The_necessities_in_Digital_Design
Young Won Lim3/22/18
Copyright (c) 2015 – 2018 Young W. Lim.
Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License".
Please send corrections (or suggestions) to [email protected].
This document was produced by using LibreOffice and Octave.
Logic (8C)K-Map
3 Young Won Lim3/22/18
Boolean Function with Maxterms
https://en.wikiversity.org/wiki/The_necessities_in_Digital_Design
Logic (8C)K-Map
4 Young Won Lim3/22/18
K-Map 3 variables (1)
2310
6754
y=1y=0
z=1z=0 z=0
x=0
x=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
3
2
0
1
7
6
4
5
x y z
x y z
x y z
x y z
x y z
x y z
x y z
x y z
2310
6754
10110100
0
1
x y zindex minterms
Logic (8C)K-Map
5 Young Won Lim3/22/18
K-Map 3 variables (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
3
2
0
1
7
6
4
5
x y z
x y z
x y z
x y z
x y z
x y z
x y z
x y z
x y
x y
x y
x y
x y z + x y z = x y ( z+ z) = x y
2310
6754
y=1y=0
z=1z=0 z=0
x=0
x=1
10110100
0
1
x y x y
x y x y
index minterms
x y z
21
a group of 2 minterms
Logic (8C)K-Map
6 Young Won Lim3/22/18
K-Map 3 variables (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
3
2
0
1
7
6
4
5
x y z
x y z
x y z
x y z
x y z
x y z
x y z
x y z
x z
x z
x z
x z
x y z + x y z = x z( y+ y) = x z
2310
6754
y=1y=0
z=1z=0 z=0
x=0
x=1
10110100
0
1
x z x z
x z x z
index minterms
x y z
21
a group of 2 minterms
x z
x z
Logic (8C)K-Map
7 Young Won Lim3/22/18
K-Map 3 variables (4)
0 1 1
0 1 0
0 0 1
0 0 0
1 1 1
1 1 0
1 0 1
1 0 0
3
2
0
1
7
6
4
5
x y z
x y z
x y z
x y z
x y z
x y z
x y z
x y z
y z
y z
y z
y z
x y z + x y z = y z ( x+x ) = y z
2310
6754
y=1y=0
z=1z=0 z=0
x=0
x=1
10110100
0
1
y z y z y z y z
index minterms
x y z
21
a group of 2 minterms
Logic (8C)K-Map
8 Young Won Lim3/22/18
K-Map 3 variables (5)
0 1 1
0 1 0
0 0 1
0 0 0
1 1 1
1 1 0
1 0 1
1 0 0
3
2
0
1
7
6
4
5
x y z
x y z
x y z
x y z
x y z
x y z
x y z
x y z
x
x
2310
6754
y=1y=0
z=1z=0 z=0
x=0
x=1
10110100
0
1 x
x
index minterms
x y z
22
a group of 4 minterms
Logic (8C)K-Map
9 Young Won Lim3/22/18
K-Map 3 variables (5)
0 1 1
0 1 0
0 0 1
0 0 0
1 1 1
1 1 0
1 0 1
1 0 0
3
2
0
1
7
6
4
5
x y z
x y z
x y z
x y z
x y z
x y z
x y z
x y z
y
y
2310
6754
y=1y=0
z=1z=0 z=0
x=0
x=1
10110100
0
1
y y
index minterms
x y z
22
a group of 4 minterms
Logic (8C)K-Map
10 Young Won Lim3/22/18
K-Map 3 variables (5)
0 1 1
0 1 0
0 0 1
0 0 0
1 1 1
1 1 0
1 0 1
1 0 0
3
2
0
1
7
6
4
5
x y z
x y z
x y z
x y z
x y z
x y z
x y z
x y z
z
z
2310
6754
y=1y=0
z=1z=0 z=0
x=0
x=1
10110100
0
1 z z
index minterms
x y z
22
a group of 4 minterms
Logic (8C)K-Map
11 Young Won Lim3/22/18
K-Map, minterms, and 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
3
2
0
1
7
6
4
5
x y z
x y z
x y z
x y z
x y z
x y z
x y z
x y z
2310
6754
10110100
0
1
x y zindex minterms
m0
m1 m
2m
3
m4
m5
m6
m7
2310
6754
10110100
0
1
M0
M1
M2
M3
M4
M5
M6
M7
x y z
Each rectangle is associated with a minterm or a maxterm which represents a particular input variable conditions.
In this table, output function value is overlaid
Logic (8C)K-Map
12 Young Won Lim3/22/18
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
1
1
1
F
2310
6754
x
y
z
1 1
1F = x z + x y z
F=1 when xz = 1
F=1 when xyz = 1
2310
6754
10110100
0
1
m0
m1 m
2m
3
m4
m5
m6
m7
an overlaid table
a simplified function
Logic (8C)K-Map
13 Young Won Lim3/22/18
Boolean Function with Maxterms
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
x y z
3
2
0
1
7
6
4
5
F0
0
0
0
0
F
0
2310
6754
0 0
0 0 0F = ( x+ z)( y+ z)(x+ z)
x
y
z
F=0 when x+z=0
F=0 when y+z=0F=0 when x+z=0
2310
6754
M0
M1
M2
M3
M4
M5
M6
M7
an overlaid table
a simplified function
Logic (8C)K-Map
15 Young Won Lim3/22/18
K-Map 4 variables (1)
0 0 1 1
0 0 1 0
0 0 0 1
0 0 0 0
0 1 1 1
0 1 1 0
0 1 0 1
0 1 0 0
1 0 1 1
1 0 1 0
1 0 0 1
1 0 0 0
1 1 1 1
1 1 1 0
1 1 0 1
1 1 0 0
3
2
0
1
7
6
4
5
11
10
8
9
15
14
12
131000 1001 1011 1010
1100 1101 1111 1110
0100 0101 0111 0110
0000 0001 0011 0010
2310
6754
101198
14151312
x y z w
x y z w
x y z w
x y zw
x y z w
x y z w
x y z w
x y zw
x y z w
x y z w
x y z w
x y zw
x y z w
x y z w
x y z w
x y zw
z=1z=0
w=1w=0 w=0
x=1
x=0
y=1
y=0
y=0
10110100
00
01
10
11
index minterms
Logic (8C)K-Map
16 Young Won Lim3/22/18
K-Map 4 variables (2)
0 0 1 1
0 0 1 0
0 0 0 1
0 0 0 0
0 1 1 1
0 1 1 0
0 1 0 1
0 1 0 0
1 0 1 1
1 0 1 0
1 0 0 1
1 0 0 0
1 1 1 1
1 1 1 0
1 1 0 1
1 1 0 0
3
2
0
1
7
6
4
5
11
10
8
9
15
14
12
131000 1001 1011 1010
1100 1101 1111 1110
0100 0101 0111 0110
0000 0001 0011 0010
2310
6754
101198
14151312
x y z w
x y z w
x y z w
x y zw
x y z w
x y z w
x y z w
x y zw
x y z w
x y z w
x y z w
x y zw
x y z w
x y z w
x y z w
x y zw
z=1z=0
w=1w=0 w=0
x=1
x=0
y=1
y=0
y=0
index minterms
Young Won Lim3/23/18
Copyright (c) 2017 - 2 018 Young W. Lim.
Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License,Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License".
Please send corrections (or suggestions) to [email protected].
This document was produced by using LibreOffice and Octave.
Set Operations (1A) 15 Young Won Lim3/23/18
Power Set Example
https://en.wikipedia.org/wiki/Power_set
Young Won Lim3/22/18
Copyright (c) 2015 - 2018 Young W. Lim.
Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License".
Please send corrections (or suggestions) to [email protected].
This document was produced by using LibreOffice and Octave.
Determinant (3A) 18Young Won Lim
03/22/2018
Gauss-Jordan Elimination
+ 2 + 1 −1
−3 −1 + 2
−2 + 1 + 2
x1
x2
x3
= + 8
= −11
= −3
x1
x2
x3
=
=
=
1 0 0
0 1 0
0 0 1
*
*
*
×c
×c+ + + +
https://en.wikiversity.org/wiki/Linear_Algebra_in_plain_view
Determinant (3A) 19Young Won Lim
03/22/2018
Gauss-Jordan Elimination – Step 1
+ 2 x1+ x
2− x
3= 8
−3 x1− x
2+ 2 x
3= −11
−2 x1+ x
2+ 2 x
3= −3
(L1)
(L2)
(L3)
+ 1 x1+ 1
2x
2− 1
2x
3= 4
−3 x1− x
2+ 2 x
3= −11
−2 x1+ x
2+ 2 x
3= −3
( 1
2× L
1)
(L2)
(L3)
+2
–3
–2
+1
–1
+1
–1
+2
+2
+8
–11
–3
+1
–3
–2
+1/2
–1
+1
–1/2
+2
+2
+4
–11
–3
+2/2 +1/2 –1/2 +8/2+ 1 x1+ 1
2x
2− 1
2x
3= 4 ( 1
2× L
1)
https://en.wikiversity.org/wiki/Linear_Algebra_in_plain_view
Determinant (3A) 20Young Won Lim
03/22/2018
Gauss-Jordan Elimination – Step 2
(L1)
(L2)
(L3)
+ 1 x1+ 1
2x
2− 1
2x
3= + 4
−3 x1− x
2+ 2 x
3= −11
−2 x1+ x
2+ 2 x
3= −3
(L1)
(3 × L1+ L
2)
(2 × L1+ L
3)
+1
–3
–2
+1/2
–1
+1
–1/2
+2
+2
+4
–11
–3
+1
0
0
+1/2
+1/2
+2
–1/2
+1/2
+1
+4
+1
+5
+ 1 x1+ 1
2x
2− 1
2x
3= + 4
0 x1+ 1
2x
2+ 1
2x
3= + 1
0 x1+ 2 x
2+ 1 x
3= + 5
(3 × L1)+ 3 x
1+ 3
2x
2− 3
2x
3= + 12
(L2)−3 x
1− x
2+ 2 x
3= −11
(2 × L1)+ 2 x
1+ 2
2x
2− 2
2x
3= + 8
(L3)−2 x
1+ x
2+ 2 x
3= −3
+3 +3/2 –3/2 +12
–3 –1 +2 –11
+2 +2/2 –2/2 +8
–2 +1 +2 –3
https://en.wikiversity.org/wiki/Linear_Algebra_in_plain_view
Determinant (3A) 21Young Won Lim
03/22/2018
Gauss-Jordan Elimination – Step 3
(L1)
(2 × L2)
(L3)
+1
0
0
+1/2
+1
+2
–1/2
+1
+1
+4
+2
+5
+ 1 x1+ 1
2x
2− 1
2x
3= + 4
0 x1+ 1 x
2+ 1 x
3= + 2
0 x1+ 2 x
2+ 1 x
3= + 5
+1
0
0
+1/2
+1/2
+2
–1/2
+1/2
+1
+4
+1
+5
+ 1 x1+ 1
2x
2− 1
2x
3= + 4
0 x1+ 1
2x
2+ 1
2x
3= + 1
0 x1+ 2 x
2+ 1 x
3= + 5
(L1)
(L2)
(L3)
0 +1 +1 +20 x1+ 1 x
2+ 1 x
3= + 2 (2 × L
2)
https://en.wikiversity.org/wiki/Linear_Algebra_in_plain_view
Determinant (3A) 22Young Won Lim
03/22/2018
Gauss-Jordan Elimination – Step 4
(L2)
(−2 × L2+ L
3)
0
+1
0
+1/2
+1
0
–1/2
+1
–1
+4
+2
+1
0 x1+ 1 x
2+ 1 x
3= + 2
0 x1+ 0 x
2− 1 x
3= + 1
+1
0
0
+1/2
+1
+2
–1/2
+1
+1
+4
+2
+5
+ 1 x1+ 1
2x
2− 1
2x
3= + 4
0 x1+ 1 x
2+ 1 x
3= + 2
0 x1+ 2 x
2+ 1 x
3= + 5
(L1)
(L2)
(L3)
0 –2 –2 –40 x1− 2 x
2− 2 x
3= −4 (−2 × L
2)
0 +2 +1 +50 x1+ 2 x
2+ 1 x
3= + 5 (L
3)
0
+ 1 x1+ 1
2x
2− 1
2x
3= + 4 (L
1)
https://en.wikiversity.org/wiki/Linear_Algebra_in_plain_view
Determinant (3A) 23Young Won Lim
03/22/2018
Gauss-Jordan Elimination – Step 5
0
0
+1
0
+1
–1
+2
+1
0 x1+ 1 x
2+ 1 x
3= + 2
0 x1+ 0 x
2− 1 x
3= + 1 0
(L2)
(L3)
0 0 +1 –10 x1− 0 x
2+ 1 x
3= −1 0(−1 × L
3)
0
0
+1
0
+1
+1
+2
–1
0 x1+ 1 x
2+ 1 x
3= + 2
0 x1+ 0 x
2+ 1 x
3= −1 0
(L2)
(−1 × L3)
+1 +1/2 –1/2 +4+ 1 x1+ 1
2x
2− 1
2x
3= + 4 (L
1)
+1 +1/2 –1/2 +4+ 1 x1+ 1
2x
2− 1
2x
3= + 4 (L
1)
https://en.wikiversity.org/wiki/Linear_Algebra_in_plain_view
Determinant (3A) 24Young Won Lim
03/22/2018
+1
Forward Phase
+2
–3
–2
+1
–1
+1
–1
+2
+2
+8
–11
–3
+1
–3
–2
+1/2
–1
+1
–1/2
+2
+2
+4
–11
–3
+1
0
0
+1/2
+1/2
+2
–1/2
+1/2
+1
+4
+1
+5
+1
0
0
+1/2
+1
+2
–1/2
+1
+1
+4
+2
+5
0
+1
0
+1/2
+1
0
–1/2
–1
+4
+2
+10
0
0
+1
0
+1
+1
+2
–10
+1 +1/2 –1/2 +4
Forward Phase - Gaussian Elimination
https://en.wikiversity.org/wiki/Linear_Algebra_in_plain_view
Determinant (3A) 25Young Won Lim
03/22/2018
Gauss-Jordan Elimination – Step 6
0
+1
0
+1/2
+1
0
0
0
+1
+7/2
+3
–1
+1 x1+ 1
2x
2+ 0 x
3= + 7
2
0 x1+ 1 x
2+ 0 x
3= + 3
0 x1+ 0 x
2+ 1 x
3= −1 0
(+ 1
2× L
3+ L
1)
(−1 × L3+ L
2)
(L3)
0
0
+1
0
+1
+1
+2
–1
0 x1+ 1 x
2+ 1 x
3= + 2
0 x1+ 0 x
2+ 1 x
3= −1 0
(L2)
(L3)
0 0 +1/2 –1/20 x1+ 0 x
2+ 1
2x
3= − 1
20(+ 1
2× L
3)
+1 +1/2 –1/2 +4(L1)
0 0 –1 +10 x1+ 0 x
2− 1 x
3= + 1 0(−1 × L
3)
0 +1 +1 +20 x1+ 1 x
2+ 1 x
3= + 2 (L
2)
+1 +1/2 –1/2 +4+ 1 x1+ 1
2x
2− 1
2x
3= + 4 (L
1)
+ 1 x1+ 1
2x
2− 1
2x
3= + 4
https://en.wikiversity.org/wiki/Linear_Algebra_in_plain_view
Determinant (3A) 26Young Won Lim
03/22/2018
Gauss-Jordan Elimination – Step 7
0
+1
0
0
+1
0
0
0
+1
+2
+3
–1
+ 1 x1+ 0 x
2− 0 x
3= + 2
0 x1+ 1 x
2+ 0 x
3= + 3
0 x1+ 0 x
2+ 1 x
3= −1 0
(− 1
2× L
2+ L
1)
(L2)
(L3)
0
+1
0
+1/2
+1
0
0
0
+1
+7/2
+3
–1
+1 x1+ 1
2x
2+ 0 x
3= + 7
2
0 x1+ 1 x
2+ 0 x
3= + 3
0 x1+ 0 x
2+ 1 x
3= −1 0
(L2)
(L3)
(L1)
0 –1/2 0 –3/20 x1− 1
2x
2+ 0 x
3= − 3
2(−1
2× L
2)
+1 +1/2 0 +7/2+ 1 x1+ 0 x
2− 0 x
3= + 2 (L
1)
https://en.wikiversity.org/wiki/Linear_Algebra_in_plain_view
Determinant (3A) 27Young Won Lim
03/22/2018
Backward Phase
0
+1
0
+1/2
+1
0
0
0
+1
+7/2
+3
–10
0
0
+1
0
+1
+1
+2
–10
+1 +1/2 –1/2 +4
0
+1
0
0
+1
0
0
0
+1
+2
+3
–10
https://en.wikiversity.org/wiki/Linear_Algebra_in_plain_view
Determinant (3A) 28Young Won Lim
03/22/2018
Gauss-Jordan Elimination
+1
+2
–3
–2
+1
–1
+1
–1
+2
+2
+8
–11–3
+1
–3
–2
+1/2
–1
+1
–1/2
+2
+2
+4
–11–3
+1
0
0
+1/2
+1/2
+2
–1/2
+1/2
+1
+4
+1
+5
+1
0
0
+1/2
+1
+2
–1/2
+1
+1
+4
+2
+5
0
+1
0
+1/2
+1
0
–1/2
–1
+4
+2
+10
0
0
+1
0
+1
+1
+2
–10
+1 +1/2 –1/2 +4
0
+1
0
+1/2
+1
0
0
0
+1
+7/2
+3
–10
0
0
+1
0
+1
+1
+2
–1
+1 +1/2 –1/2 +4
0
+1
0
0
+1
0
0
0
+1
+2
+3
–10
Backward Phase – Guass-Jordan Elimination
Forward Phase – Gaussian Elimination
https://en.wikiversity.org/wiki/Linear_Algebra_in_plain_view
Young Won Lim3/22/18
Copyright (c) 2015 - 2018 Young W. Lim.
Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License".
Please send corrections (or suggestions) to [email protected].
This document was produced by using LibreOffice and Octave.
Growth Functions (2A) 3 Young Won Lim3/22/18
Small Range
Zoom Out
All are distinguishable x2+2 x+1
x2
2 x
1
A1
x2+2 x+1
for x >−0.5 x2 < x
2+2 x+1
Growth Functions (2A) 4 Young Won Lim3/22/18
Medium Range
Zoom Out More
2 x
similarx
2+2 x+1x
2
A2
for x >−0.5 x2 < x
2+2 x+1
Growth Functions (2A) 5 Young Won Lim3/22/18
Large Range
Indistinguishable x
2+2 x+1x
2
A3
for x >−0.5 x2 < x
2+2 x+1
Growth Functions (2A) 6 Young Won Lim3/22/18
Small Range, C1=2
Zoom Out
distinguishable
x2+2 x+1
2 x2
2 x
1
B1
for x > 2.414 x2+2 x+1 < 2 x
2
for x > k f (n) < C g (n)
Growth Functions (2A) 7 Young Won Lim3/22/18
Medium Range, C1=2
Zoom Out
distinguishable
x2+2 x+1
2 x2
2 x
1
B2
for x > 2.414 x2+2 x+1 < 2 x
2
for x > k f (n) < C g (n)
Growth Functions (2A) 8 Young Won Lim3/22/18
Large Range, C1=2
Zoom Out
distinguishable
x2+2 x+1
2 x2
2 x
1
B3
for x > 2.414 x2+2 x+1 < 2 x
2
for x > k f (n) < C g (n)
Growth Functions (2A) 9 Young Won Lim3/22/18
Small Range, C1=10
Zoom Out
distinguishable
x2+2 x+1
10 x2
2 x1
C1
for x > 0.462 x2+2 x+1 < 10 x
2
for x > k f (n) < C g (n)
Growth Functions (2A) 10 Young Won Lim3/22/18
Medium Range, C1=10
Zoom Out
distinguishable
x2+2 x+1
10 x2
C2
for x > 0.462 x2+2 x+1 < 10 x
2
for x > k f (n) < C g (n)
Growth Functions (2A) 11 Young Won Lim3/22/18
Large Range, C1=10
distinguishable
x2+2 x+1
10 x2
C3
for x > 0.462 x2+2 x+1 < 10 x
2
for x > k f (n) < C g (n)
Growth Functions (2A) 12 Young Won Lim3/22/18
Small Range, C1=10
Zoom Out
distinguishable
x2+2 x+1
x2
10 x
1
D1
for 0.127 < x < 7.873 x2+2 x+1 < 10 x
Growth Functions (2A) 13 Young Won Lim3/22/18
Medium Range, C1=10
Zoom Out
distinguishable x2+2 x+1
x2
D2
10 x
for x > 7.873 10 x < x2+2 x+1
for x > k C g(n) < f (n)
Growth Functions (2A) 14 Young Won Lim3/22/18
Large Range, C1=10
distinguishable
D3
x2+2 x+1
x2
10 x
for x > 7.873 10 x < x2+2 x+1
for x > k C g(n) < f (n)
Growth Functions (2A) 15 Young Won Lim3/22/18
Big Oh
Let f and g be functions (Z→R or R→R)from the set of integers or
the set of real numbersto the set of real numbers.
We say f(x) is O(g(x)) “f(x) is big-oh of g(x)”
If there are constants C and k such that
|f(x)| ≤ C|g(x)| whenever x > k.
Growth Functions (2A) 16 Young Won Lim3/22/18
Big Oh
f (n)≤ C|g(n)|
f (n) is O(g(n))
for k < n
k
f (n)
C g(n)
Growth Functions (2A) 17 Young Won Lim3/22/18
Big Omega
Let f and g be functions (Z→R or R→R)from the set of integers or
the set of real numbersto the set of real numbers.
We say f(x) is Ω(g(x)) “f(x) is big-omega of g(x)”
If there are constants C and k such that
C|g(x)| ≤ |f(x)| whenever x > k.
Growth Functions (2A) 18 Young Won Lim3/22/18
Big Oh
for k < n
k
f (n)C|g(n)|≤ f (n)
f (n) is Ω (g(n))C g(n)
Growth Functions (2A) 19 Young Won Lim3/22/18
Matrix Notation
C|g(n)|≤ f (n)
for k < n
f (n) is Ω (g(n))
f (n)≤ C|g(n)| f (n) is O(g(n))
C1|g(n)|≤ f (n)≤ C2|g(n)| f (n) is Θ (g(n))
Growth Functions (2A) 20 Young Won Lim3/22/18
Big-Oh, -Omega, -Theta Examples
for x >−0.5 1 x2 < x
2+2 x+1
for x > 2.414 x2+2 x+1 < 2 x
2
for x > 0.462 x2+2 x+1 < 10 x
2
for x > 7.873 10 x < x2+2 x+1
x2+2 x+1 is Ω (x2)
x2+2 x+1 is O (x2)
x2+2 x+1 is O (x2)
x2+2 x+1 is Ω (x )
x2+2 x+1 is Θ (x2)
Growth Functions (2A) 21 Young Won Lim3/22/18
Matrix Notation
for k < n
f (n) is Θ (g(n)) f (n) is O(g(n))
f (n) is Ω (g(n))f (n) is Θ (g(n))
f (n) is Θ (g(n)) f (n) is O(g(n))
f (n) is Ω (g(n))f (n) is Θ (g(n))
Growth Functions (2A) 22 Young Won Lim3/22/18
Big-O Examples
x2+2 x+1 is O (x2)
x2+2 x+1 is O (x3)
x2+2 x+1 is O (x4)
x2+2 x+1
2⋅x2x
3
x4
Growth Functions (2A) 23 Young Won Lim3/22/18
Big-Ω Examples
x2+2 x+1 is Ω (x )
x2+2 x+1 is Ω (√x )
x2+2 x+1 is Ω ( log x )
√x
x
x2+2 x+1
log x
Growth Functions (2A) 24 Young Won Lim3/22/18
Big-Θ Examples (1)
x2+2 x+1 is Θ (x2)
2⋅x2
1⋅x2
x2+2∗x+1
Zoom Out
Growth Functions (2A) 25 Young Won Lim3/22/18
Big-Theta Examples (2)
x2+2 x+1 is Θ (x2)
2⋅x2
1⋅x2x
2+2∗x+1