for dld

2. Overview
• Boolean Algebra
o Axioms and Theorems
o Principle of Duality
o Boolean Operator Precedence
o Boolean Function Evaluation
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3. Boolean Algebra
 A branch of mathematics that deals with
operations (AND, OR, NOT) on logical values
with binary variables.
 Based on a set of rules derived from a small
number of basic assumptions - Axioms
29-Nov-22 CSE 225: Digital Logic Design

4. Axioms
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5. Single Variable Theorems
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6. Theorems
29-Nov-22 CSE 225: Digital Logic Design
Commutative
Associative
Distributive
Absorption

7. Theorems
29-Nov-22 CSE 225: Digital Logic Design
Combining
DeMorgan’s
Theorem
Consensus

8. Proof of DeMorgan’s Theorems
29-Nov-22 CSE 225: Digital Logic Design

9. Principle of Duality
 The dual of an algebraic expression is
obtained by interchanging + and · and
interchanging 0’s and 1’s.
 The identities appear in dual pairs. When
there is only one identity on a line the identity
is self-dual, i. e., the dual expression = the
original expression.
 Unless it happens to be self-dual, the dual of
an expression does not equal the expression
itself.
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10. • F1 = (A + C) · B + 0
Dual F1 = (A · C + B) · 1 = A · C + B
• F2 = X · Y + (W + Z)
Dual F2 =
• F3 = A · B + A · C + B · C
Dual F3 =
• F4 = X · Y + Y · Z + X · Z = X · Y + X · Z
Dual F4 =
• F5 = X · (Y + Z) = X · Y + X · Z
Dual F5 =
• Are any of these functions self-dual?
Principle of Duality
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CSE 225: Digital Logic Design

11. Boolean Operator Precedence
 The order of evaluation in a Boolean expression:
1. Parentheses
2. NOT
3. AND
4. OR
 Consequence: Parentheses appear around
OR expressions
 Example: F = A(B + C)(C + D)
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CSE 225: Digital Logic Design

12. Problem 1
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13. Problem 2
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14. Problem 3
)
Z
X
(
X
Z
)
Y
X
( +
=
+
+ Y Y
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CSE 225: Digital Logic Design

15. Boolean Function Evaluation
x y z F1 F2 F3 F4
0 0 0 0 0
0 0 1 0 1
0 1 0 0 0
0 1 1 0 0
1 0 0 0 1
1 0 1 0 1
1 1 0 1 1
1 1 1 0 1
z
x
y
x
F4
x
z
y
x
z
y
x
F3
x
F2
xy
F1
+
=
+
=
=
= z
yz
+
y
+
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CSE 225: Digital Logic Design