Admission for mba

13.1
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An introduction toAn introduction to
Boolean AlgebrasBoolean Algebras
An introduction toAn introduction to
Boolean AlgebrasBoolean Algebras
Paolo PRINETTO
Politecnico di Torino (Italy)
University of Illinois at Chicago, IL (USA)
Paolo.Prinetto@polito.it
Prinetto@uic.edu
www.testgroup.polito.it
Lecture
3.1

33.1
Goal
• This lecture first provides several definitions
of Boolean Algebras, and then focuses on
some significant theorems and properties.
• It eventually introduces Boolean Expressions
and Boolean Functions.

43.1
Prerequisites
• Students are assumed to be familiar with the
fundamental concepts of:
− Algebras, as presented, for instance, in:
⋅ F.M. Brown:
“Boolean reasoning: the logic of boolean
equations,”
Kluwer Academic Publisher, Boston MA
(USA), 1990, (chapter 1, pp. 1-21)

53.1
Prerequisites (cont’d)
− Number systems and codes, as presented, for
instance, in:
⋅ E.J.McCluskey:
“Logic design principles with emphasis on
testable semicustom circuits”,
Prentice-Hall, Englewood Cliffs, NJ, USA,
1986, (chapter 1, pp. 1-28)
or

63.1
Prerequisites (cont’d)
⋅ [Haye_94] chapter 2, pp. 51-123
or
⋅ M. Mezzalama, N. Montefusco, P. Prinetto:
“Aritmetica degli elaboratori e codifica
dell’informazione”,
UTET, Torino (Italy), 1989 (in Italian),
(chapter 1, pp. 1-38).

73.1
Homework
• Prove some of the properties of Boolean
Algebras, presented in slides 39 ÷ 59.

83.1
Further readings
• Students interested in a deeper knowledge of
the arguments covered in this lecture can
refer, for instance, to:
− F.M. Brown:
“Boolean reasoning: the logic of boolean
equations,”
Kluwer Academic Publisher, Boston MA
(USA), 1990, (chapter 2, pp. 23-69 )

93.1
OutlineOutline
• Boolean Algebras Definitions
• Examples of Boolean Algebras
• Boolean Algebras properties
• Boolean Expressions
• Boolean Functions.

103.1
Boolean Algebras DefinitionsBoolean Algebras Definitions
Boolean Algebras are defined, in the literature, in
many different ways:
• definition by lattices
• definition by properties
• definition by postulates [Huntington].

113.1
Definition by lattices
A Boolean Algebra is a complemented distributive
lattice.

123.1
Definition through properties
A Boolean Algebra is an algebraic system
( B , + , · , 0 , 1 )
where:
• B is a set, called the carrier
• + and · are binary operations on B
• 0 and 1 are distinct members of B
which has the following properties:

133.1
P1: idempotent
∀ a ∈ B:
• a + a = a
• a · a = a

143.1
P2: commutative
∀ a, b ∈ B:
• a + b = b + a
• a · b = b · a

153.1
P3: associative
∀ a, b, c ∈ B:
• a + (b + c) = (a + b) + c = a + b + c
• a · (b · c) = (a · b) · c = a · b · c

163.1
P4: absorptive
∀ a, b ∈ B:
• a + (a · b) = a
• a · (a + b) = a

173.1
P5: distributive
Each operation distributes w.r.t. the other one:
a · (b + c) = a · b + a · c
a + b · c = (a + b) · (a + c)

183.1
P6: existence of the
complement
• ∀ a ∈ B, ∃ a’ ∈ B |
− a + a’ = 1
− a · a’ = 0.
The element a’ is referred to as complement of a.

193.1
Definition by postulates
A Boolean Algebra is an algebraic system
( B , + , · , 0 , 1 )
where:
• B is a set
• + and · are binary operations in B
• 0 and 1 are distinct elements in B
satisfying the following postulates:

203.1
A1: closure
∀ a, b ∈ B:
• a + b ∈ B
• a · b ∈ B

213.1
A2 : commutative
∀ a, b ∈ B:
• a + b = b + a
• a · b = b · a

223.1
A3: distributive
∀ a, b, c ∈ B:
• a · (b + c) = a · b + a · c
• a + b · c = (a + b) · (a + c)

233.1
A4: identities
∃ 0 ∈ B | ∀ a ∈ B, a + 0 = a
∃ 1 ∈ B | ∀ a ∈ B, a · 1 = a

243.1
A5: existence of the
complement
∀ a ∈ B, ∃ a’ ∈ B |
• a + a’ = 1
• a · a’ = 0.

253.1
Some definitions
• The elements of the carrier set B={0,1} are
called constants
• All the symbols that get values ∈ B are called
variables (hereinafter they will be referred to
as x1, x2, …, xn )
• A letter is a constant or a variable
• A literal is a letter or its complement.

263.1
OutlineOutline
⇒ Examples of Boolean Algebras

273.1
Examples of Boolean
Algebras
Examples of Boolean
Algebras
Let us consider some examples of Boolean
Algebras:
• the algebra of classes
• propositional algebra
• arithmetic Boolean Algebras
• binary Boolean Algebra
• quaternary Boolean Algebra.

283.1
The algebra of classes
Suppose that every set of interest is a subset of a
fixed nonempty set S.
We call
• S a universal set
• its subsets the classes of S.
The algebra of classes consists of the set 2S
(the
set of subsets of S) together with two operations
on 2S
, namely union and intersection.

293.1
This algebra satisfies the postulates for a Boolean
Algebra, provided the substitutions:
B ↔ 2S
+ ↔ ∪
· ↔ ∩
0 ↔ ∅
1 ↔ S
Thus, the algebraic system
( 2S
, ∪, ∩, ∅, S )
ia a Boolean Algebra.
The algebra of classes (cont'd)

303.1
PropositionsPropositions
A proposition is a formula which is necessarily
TRUE or FALSE (principle of the excluded third),
but cannot be both (principle of no contradiction).
As a consequence, Russell's paradox :
“this sentence is false”
is not a proposition, since if it is assumed to be
TRUE its content implies that is is FALSE, and
vice-versa.

313.1
Propositional calculus
Let:
P a set of propositional functions
F the formula which is always false (contradiction)
T the formula which is always true (tautology)
∨ the disjunction (or)
∧ the conjunction (and)
¬ the negation (not)

323.1
The system
( P, ∨ , ∧ , F , T )
is a Boolean Algebra:
• B ↔ P
• + ↔ ∨
• · ↔ ∧
• 0 ↔ F
• 1 ↔ T
Propositional calculus (cont'd)

333.1
Arithmetic Boolean Algebra
Let:
• n be the result of a product of the elements of
a set of prime numbers
• D the set of all the dividers of n
• lcm the operation that evaluates the lowest
common multiple
• GCD the operation that evaluates the Greatest
Common Divisor.

343.1
The algebraic system:
( D, lcm, GCD, 1, n )
Is a Boolean Algebra:
• B ↔ D
• + ↔ lcm
• · ↔ GCD
• 0 ↔ 1
• 1 ↔ n
Arithmetic Boolean Algebra
(cont'd)

353.1
Binary Boolean Algebra
The system
( {0,1} , + , · , 0 , 1 )
is a Boolean Algebra, provided that the two
operations + and · be defined as follows:
+ 0 1
0 0 1
1 1 1
· 0 1
0 0 0
1 0 1

363.1
Quaternary Boolean Algebra
The system
( {a,b,0,1} , + , · , 0 , 1 )
is a Boolean Algebra provided that the two
operations + and · be defined as follows:
+ 0 a b 1 · 0 a b 1
0 0 a b 1 0 0 0 0 0
a a a 1 1 a 0 a 0 a
b b 1 b 1 b 0 0 b b
1 1 1 1 1 1 0 a b 1

373.1
OutlineOutline
⇒ Boolean Algebras properties

383.1
Boolean Algebras propertiesBoolean Algebras properties
All Boolean Algebras satisfy interesting
properties.
In the following we focus on some of them,
particularly helpful on several applications.

393.1
The Stone Representation
Theorem
“Every finite Boolean Algebra is isomorphic to the
Boolean Algebra of subsets of some finite set ”.
[Stone, 1936]

403.1
Corollary
In essence, the only relevant difference among the
various Boolean Algebras is the cardinality of the
carrier.
Stone’s theorem implies that the cardinality of the
carrier of a Boolean Algebra must be a power of
2.

413.1
Consequence
Boolean Algebras can thus be represented
resorting to the most appropriate and suitable
formalisms.
E.g., Venn diagrams can replace postulates.

423.1
Duality
Every identity is transformed into another identity
by interchanging:
• + and ·
• ≤ and ≥
• the identity elements 0 and 1.

433.1
Examples
a + 1 = 1
a · 0 = 0
a + a’ b = a + b
a (a’ + b) = a b
a + (b + c) = (a + b) + c = a + b + c
a · (b · c) = (a · b) · c = a · b · c

443.1
The inclusion relation
On any Boolean Algebra an inclusion relation ( ≤ )
is defined as follows:
a ≤ b iff a · b’ = 0.

453.1
The inclusion relation is a partial order relation,
i.e., it’s:
• reflexive : a ≤ a
• antisimmetric : a ≤ b e b ≤ a ⇒ a = b
• transitive : a ≤ b e b ≤ c ⇒ a ≤ c
Properties of the inclusion
relation

463.1
The relation gets its name from the fact that, in
the algebra of classes, it is usually represented by
the symbol ⊆ :
A ⊆ B ⇔ A ∩ B’ = ∅
AABB
in the algebra of classes

473.1
In propositional calculus, inclusion relation
corresponds to logic implication:
a ≤ b ≡ a ⇒ b
in propositional calculus

483.1
The following expressions are all equivalent:
• a ≤ b
• a b’ = 0
• a’ + b = 1
• b’ ≤ a’
• a + b = b
• a b = a .
Note

493.1
Properties of inclusion
a ≤ a + b
a b ≤ a

503.1
Complement unicity
The complement of each element is unique.

513.1
(a’)’ = a
Involution

523.1
(a + b)’ = a’ · b’
(a · b)’ = a’ + b’
De Morgan’s Laws

533.1
Generalized Absorbing
a + a’ b = a + b
a (a’+ b) = a b

543.1
Consensus Theorem
a b + a’ c + b c = a b + a’ c
(a + b) (a’ + c) (b + c) = (a + b) (a’ + c)

553.1
Equality
a = b iff a’ b + a b’ = 0
Note
The formula
a’ b + a b’
appears so often in expressions that it has been
given a peculiar name: exclusive-or or exor or
modulo 2 sum.

563.1
Boole’s expansion theorem
Every Boolean function f : Bn
→ B :
f (x1, x2, …, xn)
can be expressed as:
f (x1, x2, …, xn) =
= x1’ · f (0, x2, …, xn) + x1 · f (1, x2, …, xn)
∀ (x1, x2, …, xn) ∈ B

573.1
Dual form
f (x1, x2, …, xn) =
= [ x1’ + f (0, x2, …, xn) ] · [x1 + f (1, x2, …, xn) ]
∀ (x1, x2, …, xn) ∈ B

583.1
Remark
The expansion theorem, first proved by Boole in
1954, is mostly known as Shannon Expansion.

593.1
Note
According to Stone’s theorem, Boole’s theorem
holds independently from the cardinality of the
carrier B.

603.1
Cancellation rule
The so called cancellation rule, valid in usual
arithmetic algebras, cannot be applied to Boolean
algebras.
This means, for instance, that from the expression:
x + y = x + z
you cannot deduce that
y = z.

613.1
Demonstration
x y z x+y x+z x+y = x+z y=z
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1

623.1
Demonstration
0 0 0 0 0 T T
0 0 1 0 1 F F
0 1 0 1 0 F F
0 1 1 1 1 T T
1 0 0 1 1 T T
1 0 1 1 1 TT FF
1 1 0 1 1 TT FF
1 1 1 1 1 T T

633.1
Demonstration
0 0 0 0
0 0 1 0
0 1 0 1
0 1 1 1
1 0 0 1
1 0 1 1
1 1 0 1
1 1 1 1

643.1
Demonstration
0 0 0 0 0
0 0 1 0 1
0 1 0 1 0
0 1 1 1 1
1 0 0 1 1
1 0 1 1 1
1 1 0 1 1
1 1 1 1 1

653.1
Demonstration
0 0 0 0 0 T T
0 0 1 0 1 F F
0 1 0 1 0 F F
0 1 1 1 1 T T
1 0 0 1 1 T T
1 0 1 1 1 TT FF
1 1 0 1 1 TT FF
1 1 1 1 1 T T

663.1
Demonstration
0 0 0 0 0 T T
0 0 1 0 1 F F
0 1 0 1 0 F F
0 1 1 1 1 T T
1 0 0 1 1 T T
1 0 1 1 1 TT FF
1 1 0 1 1 TT FF
1 1 1 1 1 T T

673.1
Some Boolean Algebras satisfy some peculiar
specific properties not satisfied by other Boolean
Algebras.
An example
The properties:
x + y = 1 iff x = 1 or y = 1
x · y = 0 iff x = 0 or y = 0
hold for the binary Boolean Algebra (see slide
#28), only.
Specific properties

683.1
OutlineOutline
⇒ Boolean Expressions

693.1
Boolean Expressions
Given a Boolean Algebra defined on a carrier B,
the set of Boolean expressions can be defined
specifying:
• A set of operators
• A syntax.

703.1
Boolean Expressions
A Boolean expression is a formula defined on
constants and Boolean variables, whose semantic
is still a Boolean value.

713.1
Syntax
Two syntaxes are mostly adopted:
• Infix notationInfix notation
• Prefix notation.Prefix notation.

723.1
Infix notation
• elements of B are expressions
• symbols x1, x2, …, xn are expressions
• if g and h are expressions, then:
− (g) + (h)
− (g) · (h)
− (g)’
are expressions as well
• a string is an expression iff it can be derived
by recursively applying the above rules.

733.1
Syntactic conventions
Conventionally we are used to omit most of the
parenthesis, assuming the “·” operation have a
higher priority over the “+” one.
When no ambiguity is possible, the “·” symbol is
omitted as well.
As a consequence, for instance, the expression
((a) · (b)) + (c)
Is usually written as:
a b + c

743.1
Prefix notation
Expressions are represented by functions
composition.
Examples:
U = · (x, y)
F = + (· ( x, ‘ ( y ) ), · ( ‘ ( x ), y ) )

753.1
OutlineOutline
⇒ Boolean Functions.

763.1
Boolean functions
Several definitions are possible.
We are going to see two of them:
• Analytical definition
• Recursive definition.

773.1
Boolean functions:
Analytical definition
A Boolean function of n variables is a function
f : Bn
→ B which associates each set of values
x1, x2, …, xn ∈ B with a value b ∈ B:
f ( x1, x2, …, xn ) = b.

783.1
Boolean functions:
Recursive definition
An n-variable function f : Bn
→ B is defined
recursively by the following set of rules:
1 ∀ b ∈ B, the constant function defined as
f( x1, x2, …, xn ) = b, ∀ ( x1, x2, …, xn ) ∈ Bn
is an n-variable Boolean function
2 ∀ xi ∈ { x1, x2, …, xn } the projection function,
defined as
f( x1, x2, …, xn ) = xi ∀ ( x1, x2, …, xn ) ∈ Bn
is an n-variable Boolean function

793.1
Boolean functions:
Recursive definition (cont’d)
3 If g and h are n-variable Boolean functions,
then the functions g + h, g · h, e g’, defined as
− (g + h) (x1, x2, …, xn ) =
g(x1, x2, …, xn ) + h(x1, x2, …, xn )
− (g · h) (x1, x2, …, xn ) =
g(x1, x2, …, xn ) · h(x1, x2, …, xn )
− (g’) (x1, x2, …, xn ) = (g(x1, x2, …, xn ))’
∀ xi ∈ { x1, x2, …, xn } are also n-variable
Boolean function

803.1
Boolean functions:
Recursive definition (cont’d)
4 Nothing is an n-variable Boolean function
unless its being so follows from finitely many
applications of rules 1, 2, and 3 above.

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