meljun cortes boolean algebra

8/8/2019 MELJUN CORTES Boolean Algebra

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an algebra that is applied only to binarynumbers

has a list of operations that it can use to simplifyits equations

Boolean Algebra * Property of STI

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a set of rules defined on how a set of numbers can bemanipulated

also called Huntington’s Postulates

used to derive certain theorems applicable assuming certainconditions

Huntington’s Postulates

I. There exists a set of K objects or elements, subject to anequivalence relation, denoted “= ”, which satisfies the principle ofsubstitution.

IIa. A rule of combination “+” is defined such that A + B is in K whenever both A and B are in K .

IIb. A rule of combination “·” is defined such that A · B (abbreviatedAB ) is in K whenever both A and B are in K .

IIIa. There exists an element 0 in K such that, for every A in K , A + 0 =A.

IIIb. There exists an element 1 in K such that, for every A in K, A · 1 =A.

IV. Commutative Property


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a. on: + = +

b. Multiplication: A · B = B · A

V. Distributive Lawa. A + (B · C ) = (A + B ) · (A + C )

b. A · (B + C ) = (A · B ) + (A · C )

VI. For every element A in K , there exists an element A’ such that

A · A’ = 0

and

A + A’ = 1.

VII. There are at least two elements X and Y in K such that X = Y

Source: Hill, F. and Peterson, G., “Introduction to Switching Theory and Logical Design”, pp.42-43


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There exists a set of K ob ects or elements, sub ect

to an equivalence relation, denoted “=”, whichsatisfies the principle of substitution.

defines that variables can be defined using a(Boolean) algebraic expression by using thesymbol “=”


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A rule of combination “+” is defined such that A + B

is in K whenever both A and B are in K.

defines the addition of binary numbers

whenever two binary numbers are added, theirsum is also a binary number


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A rule of combination ·” is defined such that A · B

(abbreviated AB) is in K whenever both A and Bare in K.

defines the multiplication of binary numbers

whenever two binary numbers are multiplied,their product is also a binary number


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There exists an element 0 in K such that, for ever

A in K, A + 0 = A.

defines the zero element in the binarynumbering system

whenever zero is added to any binary number,the value of the number to which zero was

added will not change


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zero is defined such that when it is added to abinary number, the binary number will retain itsvalue


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There exists an element 1 in K such that, for ever

a in K, A · 1 = A.

defines the one element in the binarynumbering system

whenever one is multiplied to any binarynumber, the value of the number to which one

was multiplied will not change


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one is defined such that when it is multiplied toa binary number, the binary number will retainits value


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Cummutative Pro ert of Addition: A + B = B + A.

reversing the order of two binary numbers inperforming addition is shown as not to effect theresultin sum


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Cummutative Pro ert of Multi lication: A · B = B ·

A.

reversing the order of two binary numbers inerformin multi lication is shown as not to


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effect the resulting product


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Distributive Pro ert of Addition: A + B · C = A +

B) · (A + C).


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Distributive Property of Multiplication: A · (B + C) =(A · B) + (A · C).


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For ever element A in K, there exists an element

A’ such that A · A’ = 0 and A + A’ = 1.

defines the complement (A’ ) of any variable Asuch that when A’ is multiplied to A, theresulting product is zero


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There are at least two elements X and Y in K such

that X ¹ Y.

defines at least two elements in K that are notequal

Assign the following expressions to variables X and Y :


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Boolean Algebra

1. valid only for a set having

(Decimal) Algebraordinary (decimal)

2. although this propertycould be derived orproven to hold true theHuntington’s postulatesdo not includeAssociative Property

3. the distributive property

of + over · is valid sinceits set have only twoelements (0 and 1)

4. Additive andMultiplicative Inversesare not defined

set having infinite numberof elements

not applicable


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. operations are alsoundefined

6. the definition of acomplement is defined

7. especially defined tohave the property ofduality


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Theorem 1 (a) X + X = X (b) X · X = X

Theorem 2 (a) X + 1 = 1 (b) X · 0 = 0

eorem =

Theorem 4 (a) X+ (Y+Z ) = (X+Y) +Z (b) X(YZ) = (XY)Z

Theorem 5 (a) (X +Y)’ = X’Y’ (b) (XY)’ = X’+Y’

Theorem 6 (a) X + XY = X (b) X(X+Y) = X

Theorem 3 is sometimes called INVOLUTION and Theorems

6a&b are forms of ABSORPTION. Other forms of absorption

will be presented in a few examples later.

the number of combinations is directly related to

the number of variables


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2# of variables = # of combinations


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Solution:


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Theorem 1(a): X + X = X

Truth table:

Theorem 1(b): X · X = X


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Truth table:


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Theorem 2(a): X + 1 = 1

Truth table:

Theorem 2(b): X · 0 = 0


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Truth table:


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Theorem 3: (X’ )’ = X Truth table:

Y = X’

Y’ = (X’)’

Y’ = X = (X’)’

Theorem 4(a): X + (Y + Z ) = (X + Y ) + Z


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Theorem 4(b): X · (Y · Z ) = (X · Y ) · Z

X · (Y · Z) = X · (Y · Z + 0) Postulate 3a

= X · (Y + 0) · (Z + 0) Postulate 5a= ((X · Y) + (X · 0)) · Z Postulate 5b

Postulate 3a

= ((X · Y) + 0) · Z Theorem 2b

= (X · Y) · Z Postulate 3a

Truth table of X · (Y · Z ):


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Theorem 5(a): (X + Y )’ = X’Y’

(DeMorgan’s Theorem)

Postulate 6A · A’ = 0

and

A + A’ = 1

A = (X + Y )

A’ = (X + Y )’

X ’Y ’ = (X + Y )’

(X + Y) · (X + Y)’ = 0 (X + Y) · X’Y’ = 0

and

X + Y + X + Y ’ = 1 X + Y + X’Y’ = 1


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(X + Y) · X’Y’ = 1

X’Y’ · (X + Y) Postulate 4b

((X’Y’) · X)+((X’Y’) · Y) Postulate 5b

((X’X) · Y’)+((YY’) · X’) Postulate 4b

(0 · Y’) + (X’ · 0) Theorem 6

0 + 0 Theorem 2b

0 Postulate 3a


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(X + Y) + X’Y’ = 0

X + Y + X’ · X + Y +Y’ Postulate 5a

((X + X’)+ Y) · (X +(Y + Y’)) Postulate 4

(1 + Y) · (X+ 1) Postulate 6

1 · 1 Theorem 2a

1 Postulate 3b

Truth table:


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Theorem 5(b): (XY )’ = X’ + Y’

(DeMorgan’s Theorem)

Postulate 6(XY) · (XY)’ = 0 (XY) · X’ + Y’ = 0

and

(XY) + (XY)’ = 1 (XY) + X’ + Y’ = 1

(XY)·(X’ + Y’) = 0

((XY) · X’)+((XY) · Y’) Postulate 5b

((X’X) · Y)+((YY’) · X) Postulate 4b

(0 · Y) + (X · 0) Theorem 6

0 + 0 Theorem 2b

0 Postulate 3a


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(XY) + (X’ + Y’) = 1

((X’+ Y’)+ X) · ((X’+ Y’)+Y) Postulate 5a

((X+X’)+ Y’) · (X’+(Y + Y’)) Postulate 4a

(1+ Y’) · (X’ + 1) Postulate 6

1 · 1 Theorem 2a

1 Postulate 3b


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Truth table:

Theorem 6(a): X + XY = X X + XY = (X · 1) + (X · Y) Postulate 3b

= X · (1 + Y) Postulate 5b

= X · 1 Theorem 2a


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=

Truth table:


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Theorem 6(b): X (X + Y ) = X

X X+Y = X · X + X · Y Postulate 5b

= X + XY Theorem 1b

= X Theorem 6a

Truth table:


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Using the Postulates and Theorems presentedpreviously, prove the following Theorems and

write down their respective truth tables:

1. X +X ’Y =X +Y Theorem6c

2. X (X ’+Y )=XY Theorem6d

3. X ’+XY =X ’+Y Theorem6e

4. X ’(X +Y )=X ’Y ’ Theorem6f


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Solution:

1. X + X ’Y =X + Y

X + X ’Y = (X+XY)+X’Y Theorem 6a

= X+Y(X+X’) Distributive Property

(Postulate 5b)

= X+Y ·1 Postulate 6

= X+Y Postulate 3

Truth table:


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2. X (X ’+Y )=XY

X (X ’+Y ) = XX’+XY Distributive Property

(Postulate 5b)

= 0+XY Postulate 6

= XY Postulate 3

Truth table:


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3. X ’ + XY =X ’+Y

X ’ + XY = (X’+X’Y )+XY Theorem 6a

= X’+Y (X’+X) Postulate 5b

= X’+Y (1) Postulate 6

= X’+Y Postulate 3

Truth table:


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4. X + X ’Y =X ’Y ’

X + X ’Y = X’X+X’Y Distributive Property

(Postulate 5b)

= 0+X’Y Postulate 6

= X’Y Postulate 3

Truth table:


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Manipulate the following Boolean expressionsto minimize the number of terms or variables

used. Follow Operator Precedence and write-out a corresponding truth table that will showthe output values for every input combination.

5. F = (A’ + B’ + C) · (AB)’

6. F = X + (Y’ + XY’)


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Solution:

5. F = (A’ + B ’ + C ) · (AB )’

= ( A’ + B ’ + C ) · ( A’ + B ’) De Morgan’sTheorem

= ( A’ + B ’) · ( C + 1) Postulate 5b

= ( A’ + B ’) · 1 Theorem 2a

= A’ + B ’ Postulate 3b

Truth table:

6. F = X + (Y ’ + XY ’)

= X + Y ’ Theorem 6a

Truth table:

meljun cortes boolean algebra

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