Module #2 Boolean Algebra & Computing Logic page/Module-2-Computing... · 2019. 10. 30. · Boolean algebra can be described in terms of a set of elements, set of operators, and a

Module #2

Boolean Algebra & Computing Logic

ISC 115 Computing Foundations

Department of Information Science

College of Computing Sciences and Engineering(CCSE)

Kuwait University Prepared by

Prof. Mostafa Abd-El-Barr

Modified by

Prof. Jehad Al Dallal

Boolean Algebra

Algebraic Representation

Boolean Function Representation

Minimization of Boolean Functions

Outline

Boolean algebra can be described in terms of a set of elements, set of operators, and a set of axioms or postulates.

Definition: (2-valued Boolean Algebra): It is a closed algebraic system containing a set of two or more elements and two binary operators “+” (OR) and “” (AND).

The closure postulate means that for every x and y in a given set, x+y and xy also belong to the same set.

Boolean Algebra

Boolean Algebra (Cont’d)

There are Five Postulates

P1: Identity

(a) x+0 = x

(b) x1=x

P2: Commutivity

(a) x+y = y+x

(b) x y=y x

P3 : Associativity

(a) x+(y+z)=(x+y)+z (b) x(y z)=(x y) z

P4 : Distributivity

(a) x+yz=(x+y) (x+z)

(b) x(y+z)=x y+x z

P5 : Complement

(a) x+ =1

(b) x =0

xx

Boolean Algebra (Cont’d))

There are Six Theorems

T1 : Idempotency (a) x+x=x

(b) xx=x

T2 : Properties of 1 and 0 (a) x+1=1

(b) x 0=0

T3 : Absorption (a) x+xy=x (a) x+ y=x+y

(b) x(x+y)=x (b) x( +y)=xy

T4 : DeMorgan’s Law (a) (b)

T5 : Consensus (a) x y+ z+y z=x y+ z (b) (x+y) ( +z) (y+z)=(x+y) ( +z)

T6 : Involution =x

yxyx )( yxyx

x

x

x x

x

x

x

Algebraic Representation

(1) Product Term: is obtained by ANDing two (or more) variables,

e.g., .,,,, etcSQRPdacZXYBCA

(2) Sum Term: is obtained by ORing two (or more) variables,

e.g., yxCBA ,

(3) Each occurrence of a variable in either true or complemented form

within a term is called a literal, e.g., has three literals. cab

(4) A product term (sum term) P is said to be included in another product

term (sum term) Q if Q has each literal that is in P.

For example, is included in ba cba

A Boolean function can be expressed either as a sum of several product

terms (sum-of-product form) or as a product of several sum terms

(product-of-sum form)

Algebraic Representation (Cont’d)

Examples:

Q(a, b, c) =

is a sum-of-product form

P(a, b, c) =

is a product-of-sum form

cbcaba

))()(( cbcaba


If none of the product terms in a function, expressed in the Sum of Product form, are included in the other product terms of the function, then the function is said to be in the Normal sum of product(SOP).

Similarly, a function is said to be in the Normal Product of Sum (POS) if none of the sum terms of the function are included in the other.

A function is said to be in the canonical SOP (CSOP) form if each of the product terms of the function contains each and every component variable either un-complemented or complemented.

Likewise, a function is said to be in the canonical POS (CPSO) form if each of the sum terms of the function contains each and every component variable either un-complemented or complemented.


Properties of the Boolean Algebra

Operator Hierarchy (NOT, AND, OR as long as there are no parentheses).

Equality of Expressions.

Duality (the dual of an expression is obtained by replacing each “+” by “”, each “” by “+”, each “1” by “0”, and each “0” by “1” and by preserving the existence of parentheses)

Boolean Function Representation

Three Possible Representations

Graphic Form (Venn Diagrams)

Tabular Form (Truth Table)

Algebraic Form (Normal and Canonical Forms)

Boolean Function Representation (Cont’d)

Graphic Form (Venn Diagrams)

A

A B

A


Tabular Form (Truth Table)

For a set of n variables, there can be up to combinations of values.

A truth table that represents an n-variable function consists of rows and (n+1) columns.

Example: Consider the 3-variable function

f (a, b, c) =

The truth table representing this function consists of 8 rows and 4 columns as follows:

cbacaba

n2

n2


a b c f

0 0 0 0

0 0 1 1

0 1 0 0

0 1 1 1

1 0 0 1

1 0 1 1

1 1 0 0

1 1 1 0


Obtaining the CSOP from the Truth Table

1. Generate a product term corresponding to each row in which the value of the function is 1.

2. In each product term, individual variables are un-complemented if the value of the variable in that row is 1 and it is complemented if the value of the variable in that row is 0.

3. The ORing of all generated product terms

constitute the CSOP form of the function.


Example

a b c f minterms

0 0 0 0

0 0 1 1

0 1 0 0

0 1 1 1

1 0 0 1

1 0 1 1

1 1 0 0

1 1 1 0

cba

bca

cba

cba


Obtaining the CPOS from the Truth Table

1. Generate a sum term corresponding to each row in which the value of the function is 0.

2. In each sum term, individual variables are

complemented if the value of the variable in that row is 1 and it is un-complemented if the value of the variable in that row is 0.

3. The ANDing of all generated sum terms constitute

the CPOS form of the function.


a b c f maxterms

0 0 0 0

0 0 1 1

0 1 0 0

0 1 1 1

1 0 0 1

1 0 1 1

1 1 0 0

1 1 1 0

Example )( cba

)( cba

)( cba

)( cba


Decimal equivalent a b c f minterms

0 0 0 0 0

1 0 0 1 1

2 0 1 0 0

3 0 1 1 1

4 1 0 0 1

5 1 0 1 1

6 1 1 0 0

7 1 1 1 0

Each term in the CSOP is called a minterm, i.e., the CSOP is represented as sum of minterms.

cba

bca

cbacba


Decimal equivalent a b c f maxterms

0 0 0 0 0

1 0 0 1 1

2 0 1 0 0

3 0 1 1 1

4 1 0 0 1

5 1 0 1 1

6 1 1 0 0

7 1 1 1 0

)( cba

)( cba

Each term in a CPOS is called a maxterm, the CPOS is represented as the product of maxterms.

)( cba

)( cba


It should be noted that (number of minterms + number of maxterms) = , where n is the number of variables in the function.

Any product term can be expanded into a minterm.

Any sum term can be expanded into a maxterm.

n2


Algebraic Method (using Postulates & Theorems)

Graphical Method (using the K-Map)

Tabular Method (using Quine-McCluskey Algorithm)

We will only consider the Graphical Method (Karnaugh-Map)


Two Variable K-Map

Three Variable K-Map

Four Variable K-Map

Minimization of Boolean Functions (Cont’d)

For an n-variable K-map, there are squares, each correspond to a minterm of the n-variable function. There are squares in the K-map, corresponding to each variable in the function.

An two adjacent squares placed horizontally or vertically (but not diagonally) correspond to minterms which differ in only a single variable

Two minterms are said to be logically adjacent when they differ in only one variable (a variable appears in true form in one minterm and complemented form in the other and all other variables appear in the same form in both terms).

In minimizing a logic function using a K-map, the number of adjacent squares that may be combined must always represent a number that is a power of 2, such as 1, 2, 4, 8, etc.

n212 n

Designing problem

Design a logical circuit for a lock that has a four binary digits and opens for combinations 0, 2, 8, 10, 12, and 14.

Documents

Module #2 Boolean Algebra & Computing Logic page/Module-2-Computing... · 2019. 10. 30. · Boolean algebra can be described in terms of a set of elements, set of operators, and a