An introduction to Boolean Algebras Paolo PRINETTO Politecnico di Torino (Italy) University of Illinois at Chicago, IL (USA) Lecture 3.1

2 3.1 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.

3 3.1 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)

4 3.1 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

5 3.1 Prerequisites (cont’d)  [Haye_94] chapter 2, pp 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).

6 3.1 Homework  Prove some of the properties of Boolean Algebras, presented in slides 39 ÷ 59.

7 3.1 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 )

8 3.1 Outline  Boolean Algebras Definitions  Examples of Boolean Algebras  Boolean Algebras properties  Boolean Expressions  Boolean Functions.

9 3.1 Boolean Algebras Definitions Boolean Algebras are defined, in the literature, in many different ways:  definition by lattices  definition by properties  definition by postulates [Huntington].

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

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:

P1: idempotent  a  B:  a + a = a  a · a = a

P2: commutative  a, b  B:  a + b = b + a  a · b = b · a

P3: associative  a, b, c  B:  a + (b + c) = (a + b) + c = a + b + c  a · (b · c) = (a · b) · c = a · b · c

P4: absorptive  a, b  B:  a + (a · b) = a  a · (a + b) = a

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)

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.

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:

A1: closure  a, b  B:  a + b  B  a · b  B

A2 : commutative  a, b  B:  a + b = b + a  a · b = b · a

A3: distributive  a, b, c  B:  a · (b + c) = a · b + a · c  a + b · c = (a + b) · (a + c)

A4: identities  0  B |  a  B, a + 0 = a  1  B |  a  B, a · 1 = a

A5: existence of the complement  a  B,  a’  B |  a + a’ = 1  a · a’ = 0.

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 x 1, x 2, , x n )  A letter is a constant or a variable  A literal is a letter or its complement.

Outline  Boolean Algebras Definitions  Examples of Boolean Algebras  Boolean Algebras properties  Boolean Expressions  Boolean Functions.

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.

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 2 S (the set of subsets of S) together with two operations on 2 S, namely union and intersection.

This algebra satisfies the postulates for a Boolean Algebra, provided the substitutions: B  2 S +  ·  0  1  S Thus, the algebraic system ( 2 S, , , , S ) ia a Boolean Algebra. The algebra of classes (cont'd)

Propositions 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.

Propositional calculus Let: Pa set of propositional functions Fthe formula which is always false (contradiction) T the formula which is always true (tautology)  the disjunction (or)  the conjunction (and)  the negation (not)

The system ( P, , , F, T ) is a Boolean Algebra:  B  P  +   ·   0  F  1  T Propositional calculus (cont'd)

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.

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)

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

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: +0ab1·0ab1 00ab aaa11a0a0a bb1b1b00bb ab1

Outline  Boolean Algebras Definitions  Examples of Boolean Algebras  Boolean Algebras properties  Boolean Expressions  Boolean Functions.

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

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

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.

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

Duality Every identity is transformed into another identity by interchanging:  + and ·  and   the identity elements 0 and 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

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

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

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’ =  A B The inclusion relation in the algebra of classes

In propositional calculus, inclusion relation corresponds to logic implication: a  b  a  b The inclusion relation in propositional calculus

The following expressions are all equivalent:  a  b  a b’ = 0  a’ + b = 1  b’  a’  a + b = b  a b = a. Note

Properties of inclusion a  a + b a b  a

Complement unicity The complement of each element is unique.

(a’)’ = a Involution

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

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

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

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.

Boole’s expansion theorem Every Boolean function f : B n   B : f (x 1, x 2, …, x n ) can be expressed as: f (x 1, x 2, …, x n ) = = x 1 ’ · f (0, x 2, …, x n ) + x 1 · f (1, x 2, …, x n )  (x 1, x 2, …, x n )  B

Dual form f (x 1, x 2, …, x n ) = = [ x 1 ’ + f (0, x 2, …, x n ) ] · [x 1 + f (1, x 2, …, x n ) ]  (x 1, x 2, …, x n )  B

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

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

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.

Demonstration xyzx+yx+zx+y = x+zy=z

Demonstration xyzx+yx+zx+y = x+zy=z 00000TT FF FF 01111TT 10011TT TF 10111TF TF 11011TF 11111TT

Demonstration xyzx+yx+zx+y = x+zy=z

Demonstration xyzx+yx+zx+y = x+zy=z

Demonstration xyzx+yx+zx+y = x+zy=z 00000TT FF FF 01111TT 10011TT TF 10111TF TF 11011TF 11111TT

Demonstration xyzx+yx+zx+y = x+zy=z 00000TT FF FF 01111TT 10011TT TF 10111TF TF 11011TF 11111TT

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

Outline  Boolean Algebras Definitions  Examples of Boolean Algebras  Boolean Algebras properties  Boolean Expressions  Boolean Functions.

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.

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

Syntax Two syntaxes are mostly adopted:  Infix notation  Prefix notation.

Infix notation  elements of B are expressions  symbols x 1, x 2, …, x n 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.

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

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

Outline  Boolean Algebras Definitions  Examples of Boolean Algebras  Boolean Algebras properties  Boolean Expressions  Boolean Functions.

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

Boolean functions: Analytical definition A Boolean function of n variables is a function f : B n  B which associates each set of values x 1, x 2, …, x n  B with a value b  B: f ( x 1, x 2, …, x n ) = b.

Boolean functions: Recursive definition An n-variable function f : B n  B is defined recursively by the following set of rules:  b  B, the constant function defined as f( x 1, x 2, …, x n ) = b,  ( x 1, x 2, …, x n )  B n is an n-variable Boolean function  x i  { x 1, x 2, …, x n } the projection function, defined as f( x 1, x 2, …, x n ) = x i  ( x 1, x 2, …, x n )  B n is an n-variable Boolean function

Boolean functions: Recursive definition (cont’d)  If g and h are n-variable Boolean functions, then the functions g + h, g · h, e g’, defined as  (g + h) (x 1, x 2, …, x n ) = g(x 1, x 2, …, x n ) + h(x 1, x 2, …, x n )  (g · h) (x 1, x 2, …, x n ) = g(x 1, x 2, …, x n ) · h(x 1, x 2, …, x n )  (g’) (x 1, x 2, …, x n ) = (g(x 1, x 2, …, x n ))’  x i  { x 1, x 2, …, x n } are also n-variable Boolean function

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