CSE20Lecture9
BooleanAlgebra:Theoremsand
Transformations
1
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Theorems and Transformations - Discrete Mathematics - Lecture Slides, Slides of Discrete Mathematics
These lecture slides are very helpful for the student of discrete mathematics. The major points in these exam paper are: Theorems and Transformations, Boolean Algebra, Involution Laws, Absorption Law, De Morgan’s Law, Switching Algebra, Multiple Valued Boolean Algebra, Identity Elements, Commutative Laws, Boolean Transform
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CSE
Lecture
Boolean
Algebra:
Theorems
and
Transformations
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Theorems
Proofs:
Postulates
P1:
a+b =
b+a,
a
b=b
a
(commutative)
P2:
a+bc =
(a+b)
(a+c)
(distributive)
a
(b+c)
a
b +
a
c
P3:
a+0=a,
a
a
(identity)
P4:
a+a’=1,
a
a’=
(complement)
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Theorem
For every pair a, b in
B
a
a’
b = a
b; a
(a’
b)
a
b Proof: a
a’
b
(a
a’)
(a
b)
(P2)
(a
b)
(P4)
a
b
(P3)
Theorems
and
Proofs
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Theorem
De
Morgan’s
Law
Theorem: For every pair a, b in set
B:
(a+b)’
a’b’, and (ab)’
a’+b’. Proof: We show that a+b and a’b’ are complementary. In other words, we show that both of the following are true
(P4):
(a+b)+(a’b’)
(a+b)(a’b’)
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Theorems:
Switching
Algebra
vs.
Multiple
Valued
Boolean
Algebra
Boolean Algebra is termed Switching Algebra when
B
When
|B|
the system is multiple valued.
Example: M = {(0, 1, 2, 3), #, &} #
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iClicker:
M
A.
Boolean algebra can have only two elements
B.
The identity elements are 0 and
a
a
a
a C. The complement of
is
D.
Two of the above E. None of the above.
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Boolean
Transform
Given a Boolean expression, we reduce the expression (#literals, #terms) using laws and theorems of Boolean algebra.
When
B={0,1},
we can use tables to visualize the operation.
The approach follows Shannon’s expansion. - The tables are organized in two dimension space and called Karnaugh maps. Docsity.com
Boolean
Transformations
Show that a’b’+ab+a’b
a’+b Proof
a’b’+ab+a’b
a’b’+(a+a’)b
P
a’b’
b
P
a’
b Theorem
Proof
a’b’+ab+a’b
a’b’+ab+a’b+a’b Theorem
a’b’
a’b +ab+a’b
P
a’(b’+b)
(a+a’)b
P
a’* +1*b
P
a’
b
P
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