- Boolean algebra was developed by George Boole in 1847 and applied to switching circuits by Claude Shannon in 1939. - It defines basic logic operations like inverse, AND, and OR and how they relate to switching circuits. Boolean expressions correspond directly to logic gate circuits. - Truth tables specify the output values of a Boolean expression for all possible combinations of its variables. Theorems like distribution and DeMorgan's laws allow simplifying expressions and optimizing equivalent circuits.

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Unit 2 Boolean Algebra
1. Developed by George Boole in 1847
2. Applied to the Design of Switching
Circuit by Claude Shannon in 1939

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2.1 Basic Operations

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Logic Design Unit 2 Boolean Algebra Sau-Hsuan Wu
 Boolean algebra: f : {0, 1}  {0, 1}
 Basic operations
 Inverse (complement)
 AND
 OR
 Inverse is denoted by (’)
 0' = 1
 1' = 0
 Inverter

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Logic Design Unit 2 Boolean Algebra Sau-Hsuan Wu
11 1
01 0
00 1
00 0
C=A·BA B
Input Output
All possible
combinations
of inputs

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Logic Design Unit 2 Boolean Algebra Sau-Hsuan Wu
11 1
11 0
10 1
00 0
C=A+BA B

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Logic Design Unit 2 Boolean Algebra Sau-Hsuan Wu
 Basic operations of switching circuits
 A switch
 A · B  Two switches in a series
 A + B  Two switches in parallel

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2.2 Boolean Expressions and Truth
Tables

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Logic Design Unit 2 Boolean Algebra Sau-Hsuan Wu
 Boolean Expressions are formed by applications of basic
operations to one or more variables or constants, e.g.
 AB '+C
 [A(C+D)] '+BE
 Priority of operators: NOT > AND > OR
 Each expression corresponds directly to a circuit of logic gates

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Logic Design Unit 2 Boolean Algebra Sau-Hsuan Wu
 A truth table specifies the values of a Boolean expression
for every possible combinations of variables in the
expression
 E.g. AB '+C
 If an expression has n-variables, the number of different
combinations of variables is 22…=2n

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2.3 Basic Theorems and Laws

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Logic Design Unit 2 Boolean Algebra Sau-Hsuan Wu
 Basic Theorems

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Logic Design Unit 2 Boolean Algebra Sau-Hsuan Wu
 Commutative, associative and distributed laws
 Commutative laws :
XY = YX X+Y = Y+X
 Associative laws :
(XY)Z = X(YZ) = XYZ
(X+Y)+Z = X+(Y+Z) = X+Y+Z

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Logic Design Unit 2 Boolean Algebra Sau-Hsuan Wu
 Distributed law
 AND operation distributes over OR:
X(Y+Z) = XY+XZ
 OR operation also distributes over AND
X+YZ = (X+Y)(X+Z)
= XX + XY + XZ + YZ
= X ( 1+ Y + Z) + YZ
= X + YZ
This distributive law does not hold for ordinary algebra

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2.4 Simplification Theorems

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Logic Design Unit 2 Boolean Algebra Sau-Hsuan Wu
 Simplifications of Boolean expressions
 Each expression corresponds to a circuit of logic gates.
Simplifying an expression leads to a simpler circuit
 Some useful theorems
 E.g. F = A(A’+B) By the second distributive law

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Logic Design Unit 2 Boolean Algebra Sau-Hsuan Wu
 Example 1
 Example 2

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2.4 Multiplying Out and Factoring

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Logic Design Unit 2 Boolean Algebra Sau-Hsuan Wu
 An expression is said to be in sum-of-products form when
all products are the products of only single variables
 E.g. : AB’+ CD’E + AC’E
 ABC’+ DEFG + H
 When multiplying out an expression, the second
distributive law should be applied first when possible
 E.g. : (A + BC)(A + D + E) = A + BC(D + E) = A + BCD + BCE

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Logic Design Unit 2 Boolean Algebra Sau-Hsuan Wu
 An expression is in product-of-sums when all sums are
the sums of single variables
 E.g. : (A+B’)(C+D’+E)(A+C’+E’)
 The second distributive law can be applied for
factorization
 E.g. :

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Logic Design Unit 2 Boolean Algebra Sau-Hsuan Wu
 Example 1

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Logic Design Unit 2 Boolean Algebra Sau-Hsuan Wu
 Two-level circuits
 Sum-of-products
 Product-of-sums

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2.5 DeMorgan’s Laws

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Logic Design Unit 2 Boolean Algebra Sau-Hsuan Wu
 DeMorgan’s Laws
 The complement of the sum is the product of the complements
(X+Y)’= X’Y’
 The complement of the product is the sum of the complements
(XY)’= X’+ Y’
 Can be verified by using a truth table
 DeMorgan’s Laws are easily generated to n variables

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Logic Design Unit 2 Boolean Algebra Sau-Hsuan Wu
 Example 1
 Example 2

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Logic Design Unit 2 Boolean Algebra Sau-Hsuan Wu

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Logic Design Unit 2 Boolean Algebra Sau-Hsuan Wu
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