2. Digital Systems
Section 3
Boolean Algebra

3. Lecture
Digital Systems
Boolean Algebra
Boolean Algebra is a mathematical system for formulating logical statements with appropriate symbols, so that logical problems can be solved algebraically (in a manner similar to ordinary algebra).
Boolean Algebra is the mathematics of digital systems.
Boolean Algebra is introduced by an english mathematician, George Boole ( ).
The widespread use of Boolean Algebra is initiated by an american mathematician, Claude Shannon ( ).
In 1937, as a Master Student, he demonstrated that electrical applications of Boolean Algebra could construct any logical numerical relationship.
He is credited with founding both digital computer and digital circuit design theory.
George Boole
credential:
bring a computer
die photo
wafer :
This can be an hidden slide. I just want to use this to do my own planning.
I have rearranged Culler’s lecture slides slightly and add more slides. This covers everything he covers in his first lecture (and more) but may
We will save the fun part, “ Levels of Organization,” at the end (so student can stay awake): I will show the internal stricture of the SS10/20.
Notes to Patterson: You may want to edit the slides in your section or add extra slides to taylor your needs.
Claude Shannon

4. Lecture
Digital Systems
Boolean Algebra
The variables in Boolean Algebra are the truth values of True (1) and False (0).
The main operations of Boolean Algebra are
The conjunction (and, ·)
The disjunction (or, +)
The negation (not, ’)
credential:
bring a computer
die photo
wafer :
This can be an hidden slide. I just want to use this to do my own planning.
I have rearranged Culler’s lecture slides slightly and add more slides. This covers everything he covers in his first lecture (and more) but may
We will save the fun part, “ Levels of Organization,” at the end (so student can stay awake): I will show the internal stricture of the SS10/20.
Notes to Patterson: You may want to edit the slides in your section or add extra slides to taylor your needs.

5. Rules of Boolean Algebra
Lecture
Digital Systems
Rules of Boolean Algebra
Rule Number
Boolean Expression
1.a
0 · 0 = 0
1.b
1 + 1 = 1
2.a
1 · 1 = 1
2.b
0 + 0 = 0
3.a
0 · 1 = 1 · 0 = 0
3.b
0 + 1 = = 1
4.a
If x = 0, then x' = 1
4.b
If x = 1, then x' = 0

6. Rules of Boolean Algebra
Lecture
Digital Systems
Rules of Boolean Algebra
Rule Number
Boolean Expression
Property
5.a
x · 0 = 0
Annulment
5.b
x + 1 = 1
6.a
x · 1 = x
Identity
6.b
x + 0 = x
7.a
x · x = x
Idempotent
7.b
x + x = x
8.a
x · x’ = 0
Complement
8.b
x + x’ = 1 9
(x’)’ = x
Double Negation

7. Rules of Boolean Algebra
Lecture
Digital Systems
Rules of Boolean Algebra
Rule Number
Boolean Expression
Property
10.a
x · y = y · x
Commutative
10.b
x + y = y + x
11.a
x · (y · z) = (x · y) · z
Associative
11.b
x + (y + z) = (x + y) + z
12.a
x · (y + z) = x · y + x · z
Distributive
12.b
x + (y · z) = (x + y) · (x + z)
13.a
x + x · y = x
Absorption
13.b
x · (x + y) = x

8. Rules of Boolean Algebra
Lecture
Digital Systems
Rules of Boolean Algebra
Rule Number
Boolean Expression
Property
14.a
x · y + x · y’ = x
Combining
14.b
(x + y ) · ( x + y’) = x
15.a
(x · y)’ = (x’ + y’)
De Morgan’s Theorem
15.b
(x + y)’ = x’ · y’
16.a
x + x’ · y = x + y
Elimination
16.b
x · (x’ + y) = x * y
17.a
x · y + x’ · z + y · z = x · y + x’ · z
Consensus
17.b
(x + y ) · (x’ + z) · (y + z) = (x + y) · (x’+ z)

9. Rules of Boolean Algebra
Lecture
Digital Systems
Rules of Boolean Algebra
Principle of Duality
“If a Boolean Expression is valid, then the dual of that function is also valid.”
A dual of a Boolean Expression is obtained by replacing all + operators with · operators, all · with +, all 1 with 0, all 0 with 1.
To reflect the Principle of Duality, all the rules (except Rule 9) are listed in pairs.
For the purpose of simplification, the · operator is frequently omitted. If two elements are written next to each other, then it implies the use of and (·).
For example:
w + x · y = (w + x) · (w + z) is equivalent to w + xy = (w + x)(w + z)

10. Lecture
Digital Systems
De Morgan’s Theorem
A key theorem in simplifying Boolean Algebra expressions is De Morgan’s Theorem. It states:
(x + y)’ = x’ · y’
(x · y)’ = x’ + y’
Prove De Morgan’s Theorem by writing its truth tables.

11. Precedence of Operations
Lecture
Digital Systems
Precedence of Operations
In order to avoid excessive use of parentheses, a convention defines the precedence of the basic operations.
It states that, in the absence of parentheses, operations in a logic expression must be performed in the order: NOT, AND, and then OR.
For example, in A · B + A’ · C’, complements (’) must be performed first, followed by AND operations (·), followed by OR operation (+).
But, in case of A ·( B + A’) · C’, parentheses must be done first, followed by complements (’), followed by OR operation (+) since it is in the parentheses, followed by AND operations (·).

12. Boolean Simplification
Lecture
Digital Systems
Boolean Simplification
Simplify F = AB’ + C’D + AB’ + C’D.
F = AB’ + C’D + AB’ + C’D
= AB’ + C’D (Rule 7)
Simplify F = ABC + ABC’ + A’C.
F = ABC + ABC’ + A’C
= AB(C+C’) + A’C (Rule 12)
= AB(1) + A’C (Rule 8)
= AB + A’C (Rule 6)

13. Method to Complement a Function
Lecture
Digital Systems
Method to Complement a Function
De Morgan’s Theorem can be used to get a complement of a function.
The steps are:
Change F to F’, or F’ to F
Change OR to AND
Change AND to OR
Complement each individual variable
Find the complement of F = AB + C’D + B’D.
F = AB + C’D + B’D
F’ = (A’ + B’) · (C + D’) · (B + D’) (Rule 15)

14. Boolean Simplification
Lecture
Digital Systems
Boolean Simplification
Simplify F = (x1 + x3) · (x1’ + x3’).
F = (x1 + x3) · (x1’ + x3’)
= x1·x1’+ x1·x3’ + x3·x1’+ x3·x3’ (Rule 12)
= x1·x3’ + x1’·x3 (Rule 8)
Simplify F = x’yz + x’yz’ +xz.
F = x’yz + x’yz’ + xz
= x’y(z + z’) + xz (Rule 12)
= x’y ·(1) + xz (Rule 8)
= x’y + xz (Rule 6)

15. Boolean Simplification
Lecture
Digital Systems
Boolean Simplification
Find the complement of F = x + yz + xz.
F’ = x’ · (y’ + z’) · (x’ + z’) A
Simplify F = x’y’ +x’y + xy.
F = x’ + y A

16. Boolean Simplification
Lecture
Digital Systems
Boolean Simplification
Simplify F = C + (B · C)’.
F = 1 A
Simplify F = (A + C)(AD + AD’) + AC + C.
F = A + C A

17. Boolean Simplification
Lecture
Digital Systems
Boolean Simplification
Prove that (A + B)(A’ + B’) = AB’ + A’B.
Prove that AC’ + B’ C’ + AC + B’C = A’B’ + AB + AB’.

18. Venn Diagram Representation
Lecture
Digital Systems
Venn Diagram Representation
Venn Diagram is a diagram representing mathematical or logical sets pictorially as circles or closed curves within an enclosing rectangle (the universal set)
Common elements of the sets are represented by the areas of overlap among the circles.

19. Venn Diagram Representation
Lecture
Digital Systems
Venn Diagram Representation
Constant 1
Constant 0 x x'
Variable x
Variable x’

20. Venn Diagram Representation
Lecture
Digital Systems
Venn Diagram Representation x y
x · y
x + y x y z
x · y’
x · y + z

21. Exercise with Venn Diagram
Lecture
Digital Systems
Exercise with Venn Diagram
Draw and give shade to the Venn diagram of x·(y+z). x y z
Draw and give shade to the Venn diagram of x·y + x’y’z. x y z

22. Exercise with Venn Diagram
Lecture
Digital Systems
Exercise with Venn Diagram
Verify xy + x’z + yz = xy + x’z by using Venn diagram. x y z
x·y x y z
x·y x y z
x’·z y x z
y·z x y z
x’·z y z x
x·y + x’·z + y·z y z x
x·y + x’·z

23. Exercise with Venn Diagram
Lecture
Digital Systems
Exercise with Venn Diagram
Using Venn diagram, prove that (A + B)(A’ + B’) = AB’ + A’B.
Using De Morgan’s theorem, find also the complement of (A + B)(A’ + B’).

24. Lecture
Digital Systems
Homework 2
Give the dual of the Boolean expression: (X + Y) · (X’ + Z’) · (Y + Z)
Find the complement of:
A’(BC’ + B’C)
xy + y’z + z’z
Using Boolean algebra, simplify this expression: AB + A(B + C) + B(B + C)
Determine the expression that defines the shaded area of the following Venn diagram. U W V
Deadline: Tuesday, 20 September 2016.