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CEC 220 Digital Circuit DesignBoolean Algebra
Friday, January 16 CEC 220 Digital Circuit Design Slide 1 of 22
Lecture Outline
Friday, January 16 CEC 220 Digital Circuit Design
• Introduction• Basic Operations: NOT, AND, OR• Representations of Boolean Expressions• Basic Boolean Theorems• Implementation of Boolean Expressions
Slide 2 of 22
Boolean AlgebraIntroduction
Friday, January 16 CEC 220 Digital Circuit Design
• Boolean Algebra• In 1849 George Boole published a scheme for the algebraic
description of logic processes
• In the 1930’s Claude Shannon used Boolean Algebra to describe circuits built with switches
• Boolean Algebra is an effective tool for describing logic circuits
Slide 3 of 22
Boolean AlgebraBoolean Logic
Friday, January 16 CEC 220 Digital Circuit Design
• Two logic levels TRUE = HIGH = 1 FALSE = LOW = 0
+5
V
0
1 0 1
T ime
Transition from logic 1 to logic 0does not take place instantaneouslyin real digital systems
+5
V
0
Logic 1
Logic 0 Intermediate values may be visiblefor an instant
Slide 4 of 22
Time
Boolean AlgebraBasic Operations: NOT, AND, OR
Friday, January 16 CEC 220 Digital Circuit Design
• Logical NOT Description:
o The output is the complement/inverse/opposite of the input
Symbolic Representation (NOT gate):
Truth Table Representation:
Boolean Description: C = NOT A
A C
A C0 11 0
A CF TT F
Slide 5 of 22
or C = A’ or C =
Boolean AlgebraBasic Operations: NOT, AND, OR
Friday, January 16 CEC 220 Digital Circuit Design
• Logical AND Description:
o The output is TRUE if and only if all the inputs are TRUE
Symbolic Representation (AND gate):
Truth Table Representation:
Boolean Description: C = A AND B
A B C0 0 00 1 01 0 01 1 1
A B CF F FF T FT F FT T T
Slide 6 of 22
or C = AB or C = A B
Boolean AlgebraBasic Operations: NOT, AND, OR
Friday, January 16 CEC 220 Digital Circuit Design
• Logical OR Description:
o The output is TRUE if any of the inputs are TRUE
Symbolic Representation (OR gate):
Truth Table Representation:
Boolean Description: C = A OR B
A B C0 0 00 1 11 0 11 1 1
A B CF F FF T TT F TT T T
Slide 7 of 22
or C = A + B
Boolean AlgebraLogic Gates and Boolean Expressions
Friday, January 16 CEC 220 Digital Circuit Design
• Derive an expression for the output of this logic circuit? Eventually we will omit the “” in the AND gate and “+” in
the OR gate
• The logic expression is a function of three variables (A, B, and C).
B’
AB’AB’+C
Slide 8 of 22
Boolean AlgebraLogic Gates and Boolean Expressions
Friday, January 16 CEC 220 Digital Circuit Design
• Derive an expression for the output of this logic circuit?
• The output is which may also be written as [A(C+D)]’+BE This expression has five variables (A, B, C, D, and E)
C+D A(C+D) A(C+D )
BE
+
Slide 9 of 22
Boolean AlgebraLogic Gates and Boolean Expressions
Friday, January 16 CEC 220 Digital Circuit Design
• Literals Each appearance of a variable or its complement in an
expression is referred to as a literal. Example:
The expression has three variables (A, B, and C) The expression has 10 literals
AB’C+A’B+A’BC’+B’C’
Slide 10 of 22
Boolean AlgebraTruth Tables of a Logic Circuit
Friday, January 16 CEC 220 Digital Circuit Design
• Determine the truth table for the output (F) of the logic circuit
A B0 00 11 01 1
A’1100
F=A’+B1101
Two variablesFour possible inputs (i.e. 2n)
Slide 11 of 22
A+C01011111
Boolean AlgebraTruth Tables of a Logic Circuit
Friday, January 16 CEC 220 Digital Circuit Design
• Determine the truth table for the output of the logic circuit A
C
BB’
B’+C
A+C
(A+C)(B’+C)
B’+C11011101
A+C B’+C (A+C)(B’+C)01011101
(A+C)(B’+C)B’11001100
Slide 12 of 22
A00001111
B00110011
C01010101
Boolean AlgebraBasic Boolean Theorems
Friday, January 16 CEC 220 Digital Circuit Design
• Basic Theorems Principle of DUALITY:
o Given any Boolean expression its DUAL expression can be obtained by:
– Replace “ • “ by “ + “ (and vice versa), also– Replace “ 0 “ by “ 1 “ (and vice versa)
Slide 13 of 22
Boolean AlgebraBasic Boolean Theorems
Friday, January 16 CEC 220 Digital Circuit Design
• Basic Theorems Operations with 0 and 1
x0 = 0
x1 = x
x 0 x00 0 01 0 0
x 1 x10 1 01 1 1
x+0 = x
x+1 = 1
x 0 x+00 0 01 0 1
x 1 x+10 1 11 1 1
Expression Dual of Expression
Slide 14 of 22
Boolean AlgebraBasic Boolean Theorems
Friday, January 16 CEC 220 Digital Circuit Design
• Idempotent Law:
• Laws of Complementarity
• Involution Law
x + x = xx x = x
Expression Dual of Expression
= 0
Expression Dual of Expression
= 1
�́�=𝑥
Slide 15 of 22
Boolean AlgebraMore Boolean Theorems
Friday, January 16 CEC 220 Digital Circuit Design
• Commutative Law
• Associative Law
• Distributive Law
x y = y x
Expression Dual of Expression
x + y = y + x
(x y) z = x (y z)
Expression Dual of Expression
(x + y) + z = x + (y + z)
x (y + z) = (x y) + (x z)
Expression Dual of Expression
x + (y z) = (x + y) (x + z)
Slide 16 of 22
Boolean AlgebraMore Boolean Theorems
Friday, January 16 CEC 220 Digital Circuit Design
• Let’s verify the Distributive Law via a truth table
y + z01110111
x (y + z) = x y + x z
x(y + z)
0
0
0
0
0
1
1
1
LHS
x y
0
0
0
0
0
0
1
1
x z
0
0
0
0
0
1
0
1
RHS
x y + x z
0
0
0
0
0
1
1
1
y + z x(y + z) x y x z x y + x z
Slide 17 of 22
x y z0 0 00 0 10 1 00 1 11 0 01 0 11 1 01 1 1
x y z0 0 00 0 10 1 00 1 11 0 01 0 11 1 01 1 1
Boolean AlgebraBoolean Algebra Examples
Friday, January 16 CEC 220 Digital Circuit Design
• Examples Prove the following algebraically
𝑋 (𝑋+𝑌 )=𝑋𝑌𝑋 (𝑋+𝑌 ) Distributive Law
¿0+𝑋𝑌LHS
¿ 𝑋𝑌Complementarity Law
Operations with 0 and 1
¿ 𝑋 𝑋+𝑋𝑌
𝑋+𝑋𝑌=𝑋LHS 𝑋+𝑋𝑌
¿ 𝑋 (1+𝑌 )¿ 𝑋 1+𝑋𝑌 Operations with 0 and 1
Distributive Law¿ 𝑋 1 Operations with 0 and 1¿ 𝑋
Slide 18 of 22
Boolean AlgebraBoolean Algebra Examples
Friday, January 16 CEC 220 Digital Circuit Design
• Examples:
( 𝑋+𝑌 ) ( 𝑋+𝑍 )=𝑋+𝑌𝑍LHS ( 𝑋+𝑌 ) ( 𝑋+𝑍 )¿ 𝑋 (𝑋+𝑍 )+𝑌 (𝑋+𝑍 )
Last example: X+XZ = X
Distributive Law (Dual)
Idempotent Law
Distributive Law¿ 𝑋𝑋+𝑋𝑍+𝑌𝑋+𝑌𝑍¿ 𝑋+𝑋𝑍+ 𝑋𝑌+𝑌𝑍¿ 𝑋+𝑋𝑌 +𝑌𝑍¿ 𝑋+𝑌𝑍 Last example: X+XY = X
Slide 19 of 22
𝑋+𝑋𝑍+𝑋𝑌 +𝑌𝑍=𝑋 (1+𝑍 +𝑌 )+𝑌𝑍¿ 𝑋 (1)+𝑌𝑍
¿ 𝑋+𝑌𝑍
OR
Boolean AlgebraA Circuit Example
Friday, January 16 CEC 220 Digital Circuit Design
• Determine the Output of the Following Circuit
• Design a Simpler Circuit with the Same Output
𝑨
𝑩
𝑨𝑨
𝑨𝑩
𝑨𝑨+𝑨𝑩
F=𝐴 𝐴+ 𝐴𝐵¿ 𝐴+𝐴𝐵¿ ( 𝐴+𝐴 ) ( 𝐴+𝐵 )
x + (y z) = (x + y) (x + z)
Distributive Law (Dual)
¿1 ( 𝐴+𝐵 )¿ 𝐴+𝐵
A
B
A’
B’
A’+B’
Slide 20 of 22
Boolean AlgebraAn Inverter
Friday, January 16 CEC 220 Digital Circuit Design
• Implementation of an inverter
A simple RTL logicinverter
Interpret voltages per the TTL standard:
• 0 to 0.8 volts = Boolean 0 (Low)• 2.2 to 5.0 volts = Boolean 1 (High)
Vout = NOT VinVi
n Lo
w
Vin
Hig
hSlide 21 of 22
Next Lecture
Friday, January 16 CEC 220 Digital Circuit Design
• DeMorgan’s Laws• Simplification Theorems
Slide 22 of 22