Download 3 -- boolean algebra

‫طراحی مدارهای منطقی‬
‫دانشگاه آزاد اسالمی واحد پرند‬
‫نیمسال دوم ‪93-92‬‬
‫طراحی مدارهای منطقی‬
‫دانشگاه آزاد اسالمی واحد پرند‬
‫جبر بول‬
Boolean Algebra
 Boolean Algebra  Basic mathematics needed for
the study of the logic design of digital systems
 George Boole developed Boolean algebra in 1847
 Solve problems in mathematics
 Claude Shannon first applied Boolean algebra to the
design of switching circuits in 1939
Boolean Algebra
 Boolean Variable
 Such as X or Y
 Boolean Value or Constants
 0,1
 Basic Operations
 AND, OR, and complement (or inverse)
Boolean Algebra
 Basic Operations
 AND, OR, and complement (inverse)
 Complementation (Inversion)
Boolean Algebra
 Basic Operations
 AND, OR, and complement (inverse)
 AND
Boolean Algebra
 Basic Operations
 AND, OR, and complement (inverse)
 OR
Boolean Expressions and
Truth Table
Boolean expressions
 Formed by application of the basic operations to
one or more variables or constants
Boolean Expressions and
Truth Table
Boolean expressions
 Evaluation
Boolean Expressions and
Truth Table
Truth table (also called a table of combinations)
 Specifies the values of a Boolean expression for every
possible combination of values of the variables in the
expression
 2n rows for n input variables
Basic Theorems
 Involve single variable
Commutative, Associative
and Distributive laws
 Commutative )‫(جا به جایی‬
XY = YX
X+Y = Y+X
 Associative )‫(شرکت پذیری‬
(XY)Z = X(YZ) = XYZ
(X+Y)+Z = X+(Y+Z) = X+Y+Z
 Distributive )‫(توزیعی‬
X(Y+Z) = XY + XZ
X + YZ = (X+Y)(X+Z)
Logic Optimization
C
F=A’ + B•C’ + A’•B’
A
F
B
C
G=A’ + B•C’
B
A
G
Simplification Theorems
Multiplying out and
Factoring
 Multiplying out
• Forming SOP  Sum Of Products
 Factoring
• Forming POS  Products Of Sum
DeMorgan’s Law
• DeMorgan’s Laws
• Proof
• Generalized Laws
DeMorgan’s Law
• DeMorgan’s Laws
• Example
Dual
• Replacing AND with OR, OR with AND
• Replacing 0 with 1, 1 with 0
• Variables and complements are left unchanged
Exclusive-OR  XOR
Exclusive-OR  XOR
• Theorems
• Proof of distribution law
Equivalence  ExclusiveNOR  XNOR
Equivalence  ExclusiveNOR  XNOR
• Example
Consensus Theorem )‫(قانون اجماع‬
• Theorem
• Proof
• Dual
Algebraic Simplification
Combining terms
• XY + XY’ = X
Eliminating terms
• X + XY = X
Eliminating literals
• X + X’Y = X+Y
Algebraic Simplification
Example
Proving Validity of an
Equation
1. Construct a truth table and evaluate both sides
2. Manipulate one side of the equation by applying
various theorems until it is identical with the other side
3. Reduce both sides of the equation independently to
the same expression
4. It is permissible to perform the same operation on
both sides of the equation provided that the operation
is reversible. For example, it is all right to complement
both sides of the equation
Proving Validity of an
Equation
Example