Download Boolean Algebra & Logic Circuits: Lecture Notes by Dr. D. J. Jackson for ECE380 and more Slides Digital Logic Design and Programming in PDF only on Docsity!
Electrical & Computer Engineering Dr. D. J. Jackson Lecture 3-
ECE380 Digital Logic
Introduction to Logic Circuits: Boolean algebra
Axioms of Boolean algebra
- Boolean algebra: based on a set of rules derived from a small number of basic assumptions ( axioms )
- 1a 0·0=
- 1b 1+1=
- 2a 1·1=
- 2b 0+0=
- 3a 0·1=1·0=
- 3b 1+0=0+1=
- 4a If x =0 then x’ =
- 4b If x =1 then x’ =
Electrical & Computer Engineering Dr. D. J. Jackson Lecture 3-
Single-Variable theorems
- From the axioms are derived some rules for dealing with single variables
- 5a x ·0=
- 5b x +1=
- 6a x ·1= x
- 6b x +0= x
- 7a x · x = x
- 7b x + x = x
- 8a x · x’ = 0
- 8b x + x’ = 1
- 9 x’’=x
- Single-variable theorems can be proven by perfect induction
- Substitute the values x =0 and x =1 into the expressions and verify using the basic axioms
Duality
- Axioms and single-variable theorems are expressed in pairs - Reflects the importance of duality
- Given any logic expression, its dual is formed by replacing all + with ·, and vice versa and replacing all 0s with 1s and vice versa - f(a,b)=a+b dual of f(a,b)=a·b - f(x)=x+0 dual of f(x)=x·
- The dual of any true statement is also true
Electrical & Computer Engineering Dr. D. J. Jackson Lecture 3-
Induction proof of x + x’ · y = x + y
- Use perfect induction to prove x + x’ · y = x + y
1 1 0 1 1
1 0 0 1 1
0 1 1 1 1
0 0 0 0 0
x y x’y x+x’y x+y
equivalent
Perfect induction example
- Use perfect induction to prove ( xy ) ’ = x’ + y’
0
1
0
1
y’
0
0
1
1
x’
1 1 1 0 0
1 0 0 1 1
0 1 0 1 1
0 0 0 1 1
x y xy ( xy ) ’ x’+y’
equivalent
Electrical & Computer Engineering Dr. D. J. Jackson Lecture 3-
Proof (algebraic manipulation)
- Prove
- (X+A)(X’+A)(A+C)(A+D)X = AX
- (X+A)(X’+A)(A+C)(A+D)X
- (X+A)(X’+A)(A+CD)X (using 12b )
- (X+A)(X’+A)(A+CD)X
- (A)(A+CD)X (using 14b )
- (A)(A+CD)X
- AX (using 13b )
Algebraic manipulation
- Algebraic manipulation can be used to simplify Boolean expressions - Simpler expression => simpler logic circuit
- Not practical to deal with complex expressions in this way
- However, the theorems & properties provide the basis for automating the synthesis of logic circuits in CAD tools - To understand the CAD tools the designer should be aware of the fundamental concepts
Electrical & Computer Engineering Dr. D. J. Jackson Lecture 3-
Venn diagrams
X Y
Z
X
X Y X Y
(e) (f)
(g) (h)
XY X+Y
XY’ XY+Z
Y
Venn diagrams ( x + y )’= x’y’
X Y
XX Y
X Y
X (^) X’ Y’ Y
XXX Y
( X+Y ) ’ X XXX Y
X’Y’
DeMorgan’s X Theorem
Equivalent Venn diagrams imply equivalent functions
Electrical & Computer Engineering Dr. D. J. Jackson Lecture 3-
Notation and terminology
- Because of the similarity with arithmetic addition and multiplication operations, the OR and AND operations are often called the logical sum and product operations
- The expression
- ABC+A’BD+ACE’
- Is a sum of three product terms
- The expression
- (A+B+C)(A’+B+D)(A+C+E’)
- Is a product of three sum terms
Precedence of operations
- In the absence of parentheses, operations in a logical expression are performed in the order - NOT, AND, OR
- Thus in the expression AB+A’B’, the variables in the second term are complemented before being ANDed together. That term is then ORed with the ANDed combination of A and B (the AB term)
