Unit 2
Chapter 2 Boolean Algebra and Logic
Gates
 Boolean functions
 Canonical and standard forms
 Other logic operations
 Digital logic gates
 Integrated circuits
Digital Circuits, CGU
2-5 Boolean Functions
 Boolean function
• algebraic expression consists of
 binary variables (x, y, …)
 constants 0 and 1
 Logic operation symbols (AND, OR, NOT)
 For a given value of the binary variables, the function can be equal to
either 1 or 0
E.g.,
F1 = x + y’ z
F1 = 1 if x=1 or if both y’=1 and z=1
F1 = 0 otherwise
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2-5 Boolean Functions
 A Boolean function can be represented in a truth table
• A list of combinations of 1’s and 0’s assigned to the binary variables
• A column that shows the value of the function for each binary
combination
• E.g., Table 2-2 shows the truth table for the function F1 =x + y’ z
Digital Circuits, CGU
2-5 Boolean Functions
 A Boolean function can be transformed into a circuit diagram
composed of logic gates
• E.g., Fig. 2-1 – the logic-circuit diagram for F1
NOT
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AND
OR
Digital Circuits, CGU
2-5 Boolean Functions
 Boolean function => truth table (unique representation)
Boolean function => algebraic form (not unique)
• E.g.
F2 = x’y’z + x’y z + xy’
F2
0
1
0
1
1
1
0
0
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Digital Circuits, CGU
2-5 Boolean Functions
• Simplification of the function
F2 = x’y’z + x’y z + xy’ = x’z ( y’ +y ) + xy’ = x’z + xy’
Note:
(1) The circuit in (b) is simpler than the one on (a) due to
few gates, less inputs to gates, less wires, etc.
(1) By verifying truth tables, the two expressions are equivalent
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Exercise
 Draw the circuit diagram
• F=xy + x’y’
• F=xyz + x’yz + xy’z (can it be simplified?)
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2-5 Boolean Functions
 Algebraic manipulation
Digital Circuits, CGU
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2-5 Boolean Functions
 Example of simplifying the Boolean functions
Digital Circuits, CGU
2-5 Boolean Functions
 Complement of a function
• Derived algebraically through DeMorgan’s theorem
 See Table 2-1 for two variables
• DeMorgan’s theorems can be extended to three or more variables
 (A+B+C)’=(A+x)’
let B+C=x
=A’x’
by theorem 5(a)
=A’(B+C)’
substitute B+C=x
=A’(B’C’)
by theorem 5(a)
=A’B’C’
by theorem 4(b)
 (A+B+C+D+…+F)’ = A’B’C’D’…F’
(ABCD…F)’ = A’+B’+C’+D’+…+F’
• The complement of a function is obtained by interchanging AND and
OR operators and complementing each literal
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Digital Circuits, CGU
2-5 Boolean Functions
• Complement by DeMorgan’s theorems
 Example 2-2
• Complement by taking the dual of the function and complementing
each literal
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Exercises
 Find the complement
• F=xy + x’y’
• F=xyz + x’yz + xy’z
Digital Circuits, CGU
2-6 Canonical and Standard Forms
 AND-term can be called a Minterm (e.g, x’y’, x’y, xy’, xy)
 OR-term can be called a Maxterms (e.g., x+y, x’+y, …)
complement
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2-6 Canonical and Standard Forms
 Canonical form
• Boolean function can be expressed as a sum of minterms or
product of maxterms
• E.g.,
• F2:
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Exercise
Digital Circuits, CGU
2-6 Canonical and Standard Forms
 Sum of minterms
F =A+B’C = ABC +ABC’+AB’C+AB’C’+A’B’C
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2-6 Canonical and Standard Forms
 Notation for sum of minterms
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2-6 Canonical and Standard Forms
 Product of maxterms
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2-6 Canonical and Standard Forms
 Conversion between canonical forms
Sum of minterms
Product of Maxterms
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Exercise
F=  (
)
=  (
)
F’=  (
)
=  (
)
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2-6 Canonical and Standard Forms
 Standard forms
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2-6 Canonical and Standard Forms
 Nonstandard forms
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2-7 Other Logic Operations
 Functions for n binary variables
2n
2
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2-7 Other Logic Operations
Boolean functions
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2-8 Digital Logic Gates
 Implement a Boolean function with logic gates
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2-8 Digital Logic Gates
 Standard logic gates (1/2)
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2-8 Digital Logic Gates
 Standard logic gates (2/2)
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2-8 Digital Logic Gates
 Extension to multiple inputs
• A gate can be extended to have multiple inputs if the binary operation
it represents is commutative and associative
• The AND and OR operations possess these two properties
• E.g., OR
Digital Circuits, CGU
2-8 Digital Logic Gates
• The NAND and NOR are commutative, but not associative
 Their gates can be extended, but required to be modified
z
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2-8 Digital Logic Gates
• Multiple NOR and NAND gates
 Define the multiple NOR gate as a complemented OR gate
 Define the multiple NAND gate as a complemented AND gate
 In writing cascaded NOR and NAND, use the correct parentheses to
signify the proper sequence of the gates
– E.g.,
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F=[(ABC)’(DE)’]’=ABC+DE
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2-8 Digital Logic Gates
 Exclusive-OR and equivalence (XNOR) gates
• Both commutative and associative=> can be extended to more than
two inputs
• Multiple-input exclusive-OR are uncommon
 Usually constructed with other types of gates for easier implementation
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2-8 Digital Logic Gates
 Positive and negative logic
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2-8 Digital Logic Gates
 Polar indicator
• The small triangles in the inputs and output
• The presence if this polarity indicator signifies that negative logic is
used
Digital Circuits, CGU
2-9 Integrated Circuits
 An integrated circuit (IC) is a silicon semiconductor crystal,
called a chip, containing the electronic components for
constructing digital gates
 The chip is mounted in a ceramic or plastic container,
 Connections are welded to external pins
 The number of pins may range from 14 to several thousands
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2-9 Integrated Circuits
 Level of integration
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2-9 Integrated Circuits
 Digital logic families
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2-9 Integrated Circuits
 Important parameters for ICs
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2-9 Integrated Circuits
 Computer-Aided Design (CAD)