Young Won Lim3/30/18

Boolean Algebra (8A)

Young Won Lim3/30/18

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Logic (8A)Boolean Algebra 3 Young Won Lim

3/30/18

Argument

Logic (8A)Boolean Algebra 4 Young Won Lim

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Boolean Algebra

https://en.wikipedia.org/wiki/Boolean_algebra

Logic (8A)Boolean Algebra 5 Young Won Lim

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Operators

https://en.wikipedia.org/wiki/Boolean_algebra

Logic (8A)Boolean Algebra 6 Young Won Lim

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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

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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

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Digital Logic Gates

https://en.wikipedia.org/wiki/Boolean_algebra

Logic (8A)Boolean Algebra 9 Young Won Lim

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NOT Gate

https://en.wikipedia.org/wiki/Logic_gate

Logic (8A)Boolean Algebra 10 Young Won Lim

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AND, OR Gates

https://en.wikipedia.org/wiki/Logic_gate

Logic (8A)Boolean Algebra 11 Young Won Lim

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NAND, NOR Gates

https://en.wikipedia.org/wiki/Logic_gate

Logic (8A)Boolean Algebra 12 Young Won Lim

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XOR, XNOR Gates

https://en.wikipedia.org/wiki/Logic_gate

Logic (8A)Boolean Algebra 13 Young Won Lim

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CMOS Logic Gates

https://en.wikipedia.org/wiki/CMOS

Logic (8A)Boolean Algebra 14 Young Won Lim

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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

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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

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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

Young Won Lim3/30/18

References

[1] http://en.wikipedia.org/[2]