Boolean Logic

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Boolean Logic

• ITI 1121
• N. El Kadri

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What is a switching network?
Combinatorial Network A stateless network. The
output is completely determined by the values of
the input.
Sequential Network The network stores an
internal state. The output is determined by the
input, and by the internal state.
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Logic Functions Boolean Algebra
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Boolean expressions and logic circuits
Any Boolean expression can be implemented as a
logic circuit.
X A(CD)BE
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Basic Theorems Operations with 0 and 1
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Basic TheoremsIdempotent Laws
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Basic Theorems Involution Law
(X)X
B
X
CX
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Basic TheoremsLaws of Complementarity
XX 1
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Expression Simplification using the Basic
Theorems
X can be an arbitrarily complex expression.
Simplify the following boolean expressions as
much as you can using the basic theorems.
(AB D)E 1 (AB D)(AB D) (AB CD)
(CD A) (AB CD)
(AB D)E 1 1 (AB D)(AB D) 0 (AB
CD) (CD A) (AB CD) 1
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Associative Law
(XY)Z X(YZ)
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Associative Law
(XY)Z X(YZ)
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First Distributive Law
X(YZ) XYXZ
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First Distributive Law
X(YZ) XYXZ
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First Distributive Law
X(YZ) XYXZ
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First Distributive Law
X(YZ) XYXZ
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First Distributive Law
X(YZ) XYXZ
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Second Distributive Law
XYZ (XY)(XZ)
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Second Distributive Law
XYZ (XY)(XZ)
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Second Distributive Law(A different proof)
(X Y)(X Z)
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Simplification Theorems
XY XY X XY XY X(Y Y) X1 X
(X Y)(X Y) X (X Y)(X Y) XX XY
YX YY X X(Y Y)
0 X X1
X
X XY X X(1 Y) X1 X
X(X Y) X X(X Y) XX XY X1 XY
X(1 Y) X1 X
XY Y X Y (using the second distributive
law) XY Y Y XY (Y X)(Y Y)
(Y X)1 X Y
(X Y)Y XY XY YY XY 0 XY
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Examples
Simplify the following expressions
W M NP (R ST)M NP R ST
X M NP Y R ST W (X Y)(X
Y)
W XX XY YX YY
W X1 XY XY 0
W X X(Y Y) X X1 X
W M NP