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Digital Systems
Lecture 2
Dr.-Ing. Erwin Sitompul
President University
http://zitompul.wordpress.com
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Erwin Sitompul
Digital Systems 2/1
Digital Systems
Section 3
Boolean Algebra
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Erwin Sitompul
Digital Systems 2/2
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 (1815-1864).
 The widespread use of Boolean Algebra is
initiated by an american mathematician, Claude
Shannon (1916-2001).
 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
● Claude Shannon
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Erwin Sitompul
Digital Systems 2/3
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, ’)
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Erwin Sitompul
Digital Systems 2/4
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+0=1
4.a
If x = 0, then x' = 1
4.b
If x = 1, then x' = 0
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Erwin Sitompul
Digital Systems 2/5
Lecture
Digital Systems
Rules of Boolean Algebra
Rule
Number
Boolean Expression
5.a
x·0=0
5.b
x+1=1
6.a
x·1=x
6.b
x+0=x
7.a
x·x=x
7.b
x+x=x
8.a
x · x’ = 0
8.b
x + x’ = 1
9
(x’)’ = x
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Erwin Sitompul
Property
Annulment
Identity
Idempotent
Complement
Double Negation
Digital Systems 2/6
Lecture
Digital Systems
Rules of Boolean Algebra
Rule
Number
Boolean Expression
10.a
x·y=y·x
10.b
x+y=y+x
11.a
x · (y · z) = (x · y) · z
11.b
x + (y + z) = (x + y) + z
12.a
x · (y + z) = x · y + x · z
12.b
x + (y · z) = (x + y) · (x + z)
13.a
x + x·y = x
13.b
x · (x + y) = x
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Erwin Sitompul
Property
Commutative
Associative
Distributive
Absorption
Digital Systems 2/7
Lecture
Digital Systems
Rules of Boolean Algebra
Rule
Number
Boolean Expression
14.a
x · y + x · y’ = x
14.b
(x + y ) · ( x + y’) = x
15.a
(x · y)’ = (x’ + y’)
15.b
(x + y)’ = x’ · y’
16.a
x + x’ · y = x + y
16.b
x · (x’ + y) = x * y
17.a
x · y + x’ · z + y · z = x · y + x’ · z
17.b
(x + y ) · (x’ + z) · (y + z)
= (x + y) · (x’+ z)
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Erwin Sitompul
Property
Combining
De Morgan’s
Theorem
Elimination
Consensus
Digital Systems 2/8
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)
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Erwin Sitompul
Digital Systems 2/9
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.
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Erwin Sitompul
Digital Systems 2/10
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 (·).
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Erwin Sitompul
Digital Systems 2/11
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)
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Erwin Sitompul
Digital Systems 2/12
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’)
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Erwin Sitompul
(Rule 15)
Digital Systems 2/13
Lecture
Digital Systems
Boolean Simplification
Simplify F = (x1 + x3) · (x1’ + x3’).
F = (x1 + x3) · (x1’ + x3’)
= x1·x1’+ x1·x3’ + x3·x1’+ x3·x3’
= x1·x3’ + x1’·x3
(Rule 12)
(Rule 8)
Simplify F = x’yz + x’yz’ +xz.
F =
=
=
=
x’yz + x’yz’ + xz
x’y(z + z’) + xz
x’y ·(1) + xz
x’y + xz
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(Rule 12)
(Rule 8)
(Rule 6)
Erwin Sitompul
Digital Systems 2/14
Lecture
Digital Systems
Boolean Simplification
Find the complement of F = x + yz + xz.
A● F’ = x’ · (y’ + z’) · (x’ + z’)
Simplify F = x’y’ +x’y + xy.
A● F = x’ + y
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Erwin Sitompul
Digital Systems 2/15
Lecture
Digital Systems
Boolean Simplification
Simplify F = C + (B · C)’.
A● F = 1
Simplify F = (A + C)(AD + AD’) + AC + C.
A● F = A + C
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Erwin Sitompul
Digital Systems 2/16
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’.
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Erwin Sitompul
Digital Systems 2/17
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.
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Erwin Sitompul
Digital Systems 2/18
Lecture
Digital Systems
Venn Diagram Representation
Constant 1
x
Constant 0
x
x'
Variable x
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x'
Variable x’
Erwin Sitompul
Digital Systems 2/19
Lecture
Digital Systems
Venn Diagram Representation
x
y
x
x·y
y
x+y
x
x
y
y
z
x·y+z
x · y’
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Erwin Sitompul
Digital Systems 2/20
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
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Erwin Sitompul
Digital Systems 2/21
Lecture
Digital Systems
Exercise with Venn Diagram
Verify xy + x’z + yz = xy + x’z by using Venn diagram.
x
y
x
z
z
x·y
x
x·y
y
x
y
z
z
x’·z
y·z
x
y
y
x
y
z
x’·z
x
y
z
z
x·y + x’·z + y·z
x·y + x’·z
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Erwin Sitompul
Digital Systems 2/22
Lecture
Digital Systems
Exercise with Venn Diagram
a. Using Venn diagram, prove that (A + B)(A’ + B’) = AB’ + A’B.
b. Using De Morgan’s theorem, find also the complement of
(A + B)(A’ + B’).
President University
Erwin Sitompul
Digital Systems 2/23
Lecture
Digital Systems
Homework 2
1.Give the dual of the Boolean expression:
(X + Y) · (X’ + Z’) · (Y + Z)
2.Find the complement of:
a. A’(BC’ + B’C)
b. xy + y’z + z’z
3.Using Boolean algebra, simplify this expression:
AB + A(B + C) + B(B + C)
4.Determine the expression that defines the shaded area of the
following Venn diagram.
U
V
W
 Deadline: Tuesday, 29 September 2015.
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Erwin Sitompul
Digital Systems 2/24