Boolean Algebra

BOOLEAN ALGEBRABOOLEAN ALGEBRA

ICS 30/CS 30ICS 30/CS 30

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BOOLEAN ALGEBRABOOLEAN ALGEBRA Boolean Algebra, like any other deductive Boolean Algebra, like any other deductive

mathematical system, may be defined with a mathematical system, may be defined with a set of elements, a set of operators, and a set of elements, a set of operators, and a number of unproved axioms or postulates.number of unproved axioms or postulates.

A A binary operator binary operator defined on a set S of defined on a set S of elements is a rule that assigns to each pair elements is a rule that assigns to each pair of elements from S a unique element from of elements from S a unique element from S.S.


Ex., consider the relation a*b = c. We say Ex., consider the relation a*b = c. We say that * is a binary operator if it specifies a rule that * is a binary operator if it specifies a rule for finding c from the pair (a,b) and also if a, for finding c from the pair (a,b) and also if a, b, c b, c Є S. However, * is not a binary operator Є S. However, * is not a binary operator if a, b Є S, while the rule finds c Є S.if a, b Є S, while the rule finds c Є S.

The postulates of a mathematical system The postulates of a mathematical system form the basic assumptions from which it is form the basic assumptions from which it is possible to deduce the rules, theorems, and possible to deduce the rules, theorems, and properties of the system. properties of the system.



BOOLEAN ALGEBRABOOLEAN ALGEBRA The most common postulates used to The most common postulates used to

formulate various algebraic structures are:formulate various algebraic structures are: 1. Closure1. Closure. A set S is closed w/ respect to . A set S is closed w/ respect to

a binary operator if, for every pair of a binary operator if, for every pair of elements of S, the binary operator specifies elements of S, the binary operator specifies a rule for obtaining a unique element of S. a rule for obtaining a unique element of S. For example, the set of natural numbers N = For example, the set of natural numbers N = {1,2,3,4,…} is closed w/ respect to the {1,2,3,4,…} is closed w/ respect to the binary operator plus (+) by the rules of binary operator plus (+) by the rules of arithmetic addition, since for any a,b arithmetic addition, since for any a,b Є Є N we N we obtain a unique c Є N by the operation a + b obtain a unique c Є N by the operation a + b = c. = c.


The set of natural numbers is not closed w/ The set of natural numbers is not closed w/ respect to the binary operator minus (-) by respect to the binary operator minus (-) by the rules of arithmetic subtraction because the rules of arithmetic subtraction because 2-3 = -1 and 2,3 2-3 = -1 and 2,3 Є N, while (-1) Є N.Є N, while (-1) Є N.

2. Associative Law2. Associative Law. A binary operator * on . A binary operator * on a set S is said to be associative whenever: a set S is said to be associative whenever:

(x*y)*z = x*(y*z)(x*y)*z = x*(y*z) for all x,y,z for all x,y,z Є SЄ S.. 3. Commutative Law3. Commutative Law. A binary operator * . A binary operator *

on a set S is said to be commutative on a set S is said to be commutative whenever:whenever:

x*y = y*xx*y = y*x for all x,y for all x,y Є S.Є S.



BOOLEAN ALGEBRABOOLEAN ALGEBRA 4. Identity element4. Identity element. A set S is said to . A set S is said to

have an identity element w/ respect to a have an identity element w/ respect to a binary operation * on S if there exists an binary operation * on S if there exists an element e element e Є S w/ the property: Є S w/ the property: e*x = x*ee*x = x*e = x = x for every x for every x Є S.Є S.

Ex: The element 0 is an identity element w/ Ex: The element 0 is an identity element w/ respect to operation + on the set of integers respect to operation + on the set of integers I = {…,-3,-2,-1,0,1,2,3,…} since:I = {…,-3,-2,-1,0,1,2,3,…} since:xx+0 = 0++0 = 0+xx = = xx for any for any xx Є IЄ IThe set of natural numbers N has no identity The set of natural numbers N has no identity element since 0 is excluded from the set.element since 0 is excluded from the set.


BOOLEAN ALGEBRABOOLEAN ALGEBRA 5. Inverse5. Inverse. A set S having the identity . A set S having the identity

element e w/ respect to a binary operator * is element e w/ respect to a binary operator * is said to have an inverse whenever, for every said to have an inverse whenever, for every xx Є S, there exists an element Є S, there exists an element yy Є S such that: Є S such that:

x*yx*y = = ee

Ex: In the set of integers I w/ Ex: In the set of integers I w/ ee = 0, the = 0, the inverse of an element a is inverse of an element a is (-a)(-a) since since aa + + (-a)(-a) = = 0.0.

6. Distributive law6. Distributive law. If * and . Are two binary . If * and . Are two binary operators on a set S, * is said to be operators on a set S, * is said to be distributive over . Whenever:distributive over . Whenever:


x*(yx*(y••z) = (x*y)z) = (x*y)••(x*z)(x*z) An example of an algebraic structure is a An example of an algebraic structure is a

fieldfield. A field is a set of elements, together w/ . A field is a set of elements, together w/ two binary operators, each having properties two binary operators, each having properties 1 to 5 and both operators combined to give 1 to 5 and both operators combined to give property 6. The set of real numbers together property 6. The set of real numbers together w/ the binary operators + and w/ the binary operators + and • form the field • form the field of real numbers. The field of real numbers is of real numbers. The field of real numbers is the basis for arithmetic and ordinary algebra.the basis for arithmetic and ordinary algebra.




The operators and postulates have the The operators and postulates have the following meanings:following meanings:

The binary operator + defines addition.The binary operator + defines addition.The additive identity is 0.The additive identity is 0.The additive inverse defines subtraction.The additive inverse defines subtraction.The binary operator The binary operator • defines • defines multiplicationmultiplicationThe multiplicative identity is 1.The multiplicative identity is 1.The multiplicative inverse of The multiplicative inverse of aa=1/=1/aa defines division,i.e., defines division,i.e., aa.1/.1/aa =1. =1.


The only distributive law applicable is that The only distributive law applicable is that of of • and +: • and +: a • (b + c) = (a • b) + (a • c)a • (b + c) = (a • b) + (a • c)

AXIOMATIC DEFINITION AXIOMATIC DEFINITION

In 1854, George Boole introduced a In 1854, George Boole introduced a systematic treatment of logic and systematic treatment of logic and developed for this purpose an algebraic developed for this purpose an algebraic system now called system now called Boolean AlgebraBoolean Algebra..

In 1938, C.E. Shannon introduced a two-In 1938, C.E. Shannon introduced a two-valued Boolean algebra called switching valued Boolean algebra called switching algebra, in w/c he demonstrated thatalgebra, in w/c he demonstrated that



the properties of bistable electrical the properties of bistable electrical switching circuits can be represented by switching circuits can be represented by this algebra.this algebra.

For the formal definition of Boolean For the formal definition of Boolean algebra, we shall employ the postulates algebra, we shall employ the postulates formulated by E.V. Huntington in 1904. formulated by E.V. Huntington in 1904. These postulates or axioms are not These postulates or axioms are not unique for defining Boolean algebra. unique for defining Boolean algebra. Other sets of postulates have been used.Other sets of postulates have been used.



Boolean algebra is an algebraic Boolean algebra is an algebraic structure defined on a set of elements B structure defined on a set of elements B together w/ two binary operators + and together w/ two binary operators + and • • provided the provided the ff. (Huntington) postulates ff. (Huntington) postulates are satisfied:are satisfied:

1.1. (a)Closure w/ respect to the operator +.(a)Closure w/ respect to the operator +.

(b)Closure w/ respect to the operator (b)Closure w/ respect to the operator •.•.

2. (a) An identity element w/ respect to +, 2. (a) An identity element w/ respect to +, designated by 0: designated by 0: xx + 0=0 + + 0=0 + xx = = xx..



(b)An identity element w/ respect to (b)An identity element w/ respect to •, •, designated by 1: designated by 1: xx • • 11 = = 11 • x = x. • x = x.

3. (a)Commutative w/ respect to +: 3. (a)Commutative w/ respect to +: xx++yy = = yy++xx..

(b)Commutative w/ respect to (b)Commutative w/ respect to ••:: x•y = y•x x•y = y•x

4. (a) 4. (a) • • is distributive over +: is distributive over +: x•(y+z)=(x•y) + x•(y+z)=(x•y) + (x•z). (x•z).

(b)+ is distributive over •: (b)+ is distributive over •: x+(y•z) = (x+y)• x+(y•z) = (x+y)• (x+z).(x+z).



5. For every element 5. For every element xx ЄЄ BB, , there exists an there exists an element element xx Є Є BB (called the complement of (called the complement of x) such that: (a) x) such that: (a) xx++xx = 1 and (b) = 1 and (b) xx•x = 0 •x = 0

6. There exists at least two elements 6. There exists at least two elements xx,,yy Є Є BB such that such that xx = = yy..



Comparing Boolean algebra w/ arithmetic Comparing Boolean algebra w/ arithmetic and ordinary algebra (the field of real and ordinary algebra (the field of real numbers), we note the ff. differences:numbers), we note the ff. differences:

Huntington postulates do not include the Huntington postulates do not include the associative law. However, this law holds associative law. However, this law holds for Boolean algebra and can be derived for Boolean algebra and can be derived (for both operators) from the other (for both operators) from the other postulates.postulates.

The distributive law of + over The distributive law of + over •, i.e.,•, i.e., x + x + (y (y •• z) z) = = (x+y)(x+y) • (x+z), • (x+z), is valid for Boolean is valid for Boolean algebra, but not for ordinary algebra.algebra, but not for ordinary algebra.



3. Boolean algebra does not have additive or 3. Boolean algebra does not have additive or multiplicative inverses; therefore, there multiplicative inverses; therefore, there are no subtraction or division operations.are no subtraction or division operations.

4. Postulate 5 defines an operator called 4. Postulate 5 defines an operator called complementcomplement w/c is not available in w/c is not available in ordinary algebra.ordinary algebra.

5. Ordinary algebra deals w/ real numbers, 5. Ordinary algebra deals w/ real numbers, w/c constitute an infinite set of elements. w/c constitute an infinite set of elements. Boolean algebra deals w/ the as yet Boolean algebra deals w/ the as yet undefined set of elements undefined set of elements BB, but in the , but in the two-valued Boolean algebra defined two-valued Boolean algebra defined



later (and of interest in our subsequent use of later (and of interest in our subsequent use of this algebra), this algebra), BB is defined as a set w/ only two is defined as a set w/ only two elements, 0 and 1.elements, 0 and 1.

Boolean algebra resembles ordinary algebra Boolean algebra resembles ordinary algebra in some respects. The choice of symbols + in some respects. The choice of symbols + and and • is • is intentional to facilitate Boolean intentional to facilitate Boolean algebraic manipulations by persons already algebraic manipulations by persons already familiar w/ ordinary algebra. Although one can familiar w/ ordinary algebra. Although one can use some knowledge from ordinary algebra to use some knowledge from ordinary algebra to deal w/ Boolean algebra, the beginner must deal w/ Boolean algebra, the beginner must be careful not substitute the rules of be careful not substitute the rules of



ordinary algebra where they are not ordinary algebra where they are not applicable.applicable.

It is important to distinguish bet. The It is important to distinguish bet. The elements of the set of an algebraic structure elements of the set of an algebraic structure and the variables of an algebraic system. For and the variables of an algebraic system. For example, the elements of the field of real example, the elements of the field of real numbers are numbers, whereas variables numbers are numbers, whereas variables such as such as a,b,ca,b,c, etc., used in ordinary algebra, , etc., used in ordinary algebra, are symbols that stand for real numbers. are symbols that stand for real numbers. Similarly, in Boolean algebra, one defines Similarly, in Boolean algebra, one defines the elements of the set the elements of the set BB, and , and



variables such as variables such as x,y,zx,y,z are merely are merely symbols that represent the elements. At symbols that represent the elements. At this point, it is important to realize that in this point, it is important to realize that in order to have a Boolean algebra, one order to have a Boolean algebra, one must show:must show:

1.1. the elements of the set the elements of the set BB,,2.2. the rules of operation for the two binary the rules of operation for the two binary

operators, andoperators, and3.3. that the set of elements that the set of elements BB, together w/ , together w/

the two operators, satisfies the six the two operators, satisfies the six Huntington postulates.Huntington postulates.



We deal only w/ a two-valued We deal only w/ a two-valued Boolean algebra, I.e., one w/ only Boolean algebra, I.e., one w/ only two elements. Two-valued Boolean two elements. Two-valued Boolean algebra has applications in set algebra has applications in set theory (the algebra of classes) and theory (the algebra of classes) and in propositional logic. We are in propositional logic. We are interested w/ the application of interested w/ the application of Boolean algebra to gate-type Boolean algebra to gate-type circuits.circuits.



Two-Valued Boolean AlgebraTwo-Valued Boolean AlgebraIs defined on a set of two elements, Is defined on a set of two elements, BB = =

{0,1}, w/ rules for the two binary {0,1}, w/ rules for the two binary operators + and operators + and • • as shown in the ff. as shown in the ff. operator tables (the rule for the operator tables (the rule for the complement operator is for verification complement operator is for verification of postulate 5):of postulate 5):



BOOLEAN ALGEBRABOOLEAN ALGEBRA xx yy x • yx • y

00 00 00

00 11 00

11 00 00

11 11 11

xx yy x + yx + y

00 00 00

00 11 11

11 00 11

11 11 11

xx xx

00 11

11 00


These rules are exactly the same as the These rules are exactly the same as the AND, OR and NOT operations, AND, OR and NOT operations, respectively, defined in the previous respectively, defined in the previous slide. We must now show that the slide. We must now show that the Huntington postulates are valid for the Huntington postulates are valid for the set set BB = {0,1} and the two binary = {0,1} and the two binary operators defined.operators defined.

1. 1. ClosureClosure is obvious from the tables since is obvious from the tables since the result of each operation isthe result of each operation is



Either 1 or 0 and 1, 0 Either 1 or 0 and 1, 0 Є Є BB.. 2. From the tables we see that:2. From the tables we see that:

(a) 0 + 0 = 0(a) 0 + 0 = 0 0 + 1 = 1 + 0 = 10 + 1 = 1 + 0 = 1(b) 1 (b) 1 • • 1 = 11 = 1 1 1 • • 0 = 0 0 = 0 • • 11 = = 00

w/c establishes the two w/c establishes the two identity elementsidentity elements 0 0 for + and 1 for for + and 1 for • • as defined by postulate 2.as defined by postulate 2.

3. The 3. The commutativecommutative laws are obvious from laws are obvious from the symmetry of the binary operator tables.the symmetry of the binary operator tables.

4. (a) The 4. (a) The distributivedistributive law law xx • (y+z) = (x •y) + • (y+z) = (x •y) + (x •z) (x •z) can be shown to hold true from the can be shown to hold true from the operator tables by forming a truth tableoperator tables by forming a truth table



of all possible values of of all possible values of xx,,yy, and , and zz. For . For each combination, we derive x each combination, we derive x •(y+z) •(y+z) and show that the value is the same as x and show that the value is the same as x • • ((y + zy + z) and show that the value is the ) and show that the value is the same as (same as (x • yx • y) + () + (x • zx • z). ).



BOOLEAN ALGEBRABOOLEAN ALGEBRA xx y y zz yy + + zz xx • • ((y + zy + z)) x x • • yy x x • • zz ((xx••yy)+ )+

((xx••zz))

00 00 00 00 00 00 00 00

00 00 11 11 00 00 00 00

00 11 0 0 11 00 00 00 00

00 11 11 11 00 00 00 00

11 00 00 00 00 00 00 00

11 00 11 11 11 00 11 11

11 11 00 11 11 11 00 11

11 11 11 11 11 11 11 11


(b) The (b) The distributivedistributive law of + over law of + over • • can be can be shown to hold true by means of a truth shown to hold true by means of a truth table similar to the one in the previous table similar to the one in the previous slide.slide.

5. From the complement table it is easily 5. From the complement table it is easily shown that:shown that:

(a)(a)xx + + xx = 1, since 0 + 0 = 0 + 1 = 1 and 1 + = 1, since 0 + 0 = 0 + 1 = 1 and 1 + 1 = 1 + 0 = 11 = 1 + 0 = 1

(b)(b)x • x = x • x = 0, since 0 0, since 0 • • 00 = = 00 • 1• 1 = = 0 and 1 0 and 1 • • 11 = = 11 • • 0 = 0 w/c verifies postulate 5.0 = 0 w/c verifies postulate 5.



6.6. Postulate 6 is satisfied because the two-Postulate 6 is satisfied because the two-valued Boolean algebra has two distinct valued Boolean algebra has two distinct elements 1 and 0 w/ 1 = 0.elements 1 and 0 w/ 1 = 0.

We have just established a two-valued We have just established a two-valued Boolean algebra having a set of two Boolean algebra having a set of two elements, 1 and 0, two binary operators elements, 1 and 0, two binary operators w/ operation rules equivalent to the AND w/ operation rules equivalent to the AND and OR operations, and a complement and OR operations, and a complement operator equivalent to the NOT operator. operator equivalent to the NOT operator. The two-valued Boolean algebra defined The two-valued Boolean algebra defined is also called “switching algebra” by is also called “switching algebra” by engineers. engineers.



DualityDualityThe Huntington postulates have been listed The Huntington postulates have been listed

in pairs and designated by part (a) and in pairs and designated by part (a) and part (b). One part may be obtained from part (b). One part may be obtained from the other if the binary operators and the the other if the binary operators and the identity elements are interchanged. This identity elements are interchanged. This important property of Boolean algebra is important property of Boolean algebra is called the called the duality principleduality principle. It states . It states that every algebraic expression that every algebraic expression deducible from the postulates of Boolean deducible from the postulates of Boolean algebra remains valid if the operators algebra remains valid if the operators and identity elements are interchanged.and identity elements are interchanged.



In a two-valued Boolean algebra, the In a two-valued Boolean algebra, the identity elements and the elements of identity elements and the elements of the set the set BB are the same: 1 and 0. The are the same: 1 and 0. The duality principle has many applications. duality principle has many applications. If the dual of an algebraic expression is If the dual of an algebraic expression is desired, we simply interchange OR and desired, we simply interchange OR and AND operators and replace 1’s by 0’s AND operators and replace 1’s by 0’s and 0’s by 1’s.and 0’s by 1’s.



The next two slides lists the six theorems of The next two slides lists the six theorems of Boolean algebra and four of its postulates. Boolean algebra and four of its postulates. The theorems, like the postulates, are listed The theorems, like the postulates, are listed in pairs; each relation is the dual of the one in pairs; each relation is the dual of the one paired w/ it. The postulates are basic axioms paired w/ it. The postulates are basic axioms of the algebraic structure and need no proof. of the algebraic structure and need no proof. The theorems must be proven from the The theorems must be proven from the postulates. The theorems involving two or postulates. The theorems involving two or three variables may be proven algebraically three variables may be proven algebraically from the postulates and the theorems w/c from the postulates and the theorems w/c have already been proven. The theoremshave already been proven. The theorems



BOOLEAN ALGEBRABOOLEAN ALGEBRAPOSTULATES AND THEOREMS OF POSTULATES AND THEOREMS OF BOOLEAN ALGEBRABOOLEAN ALGEBRA

Postulate 2Postulate 2 (a)(a)xx +0 = +0 = xx (b)(b)xx.1 = .1 = xx

Postulate 5Postulate 5 (a)(a)x x + + xx = 1 = 1 (b)(b)xx..xx = 0 = 0

Theorem 1Theorem 1 (a)(a)x x + + xx = = xx (b)(b)xx..xx = = xx

Theorem 2Theorem 2 (a)(a)x x + 1 = 1+ 1 = 1 (b)(b)xx.0 = 0.0 = 0

Theorem 3,involutionTheorem 3,involution xx = = xx

Postulate 3,commutativePostulate 3,commutative (a)(a)xx++yy==yy++xx (b)(b)xyxy = = yxyx

Theorem 4,associativeTheorem 4,associative (a)(a)x x + (+ (y y + + zz) = ) = ((x x + + yy) + ) + zz

(b)(b)xx((yzyz) = ) = ((xyxy))zz


BOOLEAN ALGEBRABOOLEAN ALGEBRAPOSTULATES AND THEOREMS OF POSTULATES AND THEOREMS OF BOOLEAN ALGEBRABOOLEAN ALGEBRA

Postulate 4,distributivePostulate 4,distributive (a)(a)xx((yy++zz)= )= xyxy++xzxz

(b)(b)xx++yzyz= = ((xx++yy)()(xx++zz))

Theorem 5,DeMorganTheorem 5,DeMorgan (a)((a)(x x + + y)y) = = x yx y (b)((b)(xy)xy) = = x x + + yy

Theorem 6,absorptionTheorem 6,absorption (a)(a)x x + + xyxy = = xx (b)(b)x(x+y)x(x+y) = = xx


of Boolean algebra can be shown to hold of Boolean algebra can be shown to hold true by means of truth tables. In truth tables, true by means of truth tables. In truth tables, both sides of the relation are checked to both sides of the relation are checked to yield identical results for all possible yield identical results for all possible combinations of variables involved.combinations of variables involved.

Operator PrecedenceOperator Precedence(1) Parentheses (2) NOT (3) AND (4) OR(1) Parentheses (2) NOT (3) AND (4) OR


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