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Young Won Lim3/30/18
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Logic (8A)Boolean Algebra 4 Young Won Lim
3/30/18
Boolean Algebra
https://en.wikipedia.org/wiki/Boolean_algebra
Logic (8A)Boolean Algebra 5 Young Won Lim
3/30/18
Operators
https://en.wikipedia.org/wiki/Boolean_algebra
Logic (8A)Boolean Algebra 6 Young Won Lim
3/30/18
Laws (1)
https://en.wikipedia.org/wiki/Boolean_algebra
x+(y+z) = (x+y)+zx(yz) = (xy)zx+y = y+xxy = yzx(y+z) = xy + xzx+0=xx*1=xx*0=0
Logic (8A)Boolean Algebra 7 Young Won Lim
3/30/18
Laws (2)
https://en.wikipedia.org/wiki/Boolean_algebra
x+1=1
x+x=x
x*x=x
x(x+y)=x
x+xy=x
x+yz=(x+y)(x+z)
x x = 0x+ x = 1
x y =(x+ y)
x+ y = (x y)
Logic (8A)Boolean Algebra 8 Young Won Lim
3/30/18
Digital Logic Gates
https://en.wikipedia.org/wiki/Boolean_algebra
Logic (8A)Boolean Algebra 10 Young Won Lim
3/30/18
AND, OR Gates
https://en.wikipedia.org/wiki/Logic_gate
Logic (8A)Boolean Algebra 11 Young Won Lim
3/30/18
NAND, NOR Gates
https://en.wikipedia.org/wiki/Logic_gate
Logic (8A)Boolean Algebra 12 Young Won Lim
3/30/18
XOR, XNOR Gates
https://en.wikipedia.org/wiki/Logic_gate
Logic (8A)Boolean Algebra 13 Young Won Lim
3/30/18
CMOS Logic Gates
https://en.wikipedia.org/wiki/CMOS
Logic (8A)Boolean Algebra 14 Young Won Lim
3/30/18
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 Lim
3/30/18
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 Lim
3/30/18
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 Lim
3/30/18
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 Lim
3/30/18
`
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