computer programming skills revision prepared by: ghader kurdi

Computer Programming Computer Programming Skills Revision Skills Revision

Prepared by: Ghader KurdiPrepared by: Ghader Kurdi

Chapter 1Chapter 1Number SystemsNumber Systems

ContentsContents

Number Systems Number Systems Conversion Among BasesConversion Among Bases Binary Addition and MultiplicationBinary Addition and Multiplication

Number SystemsNumber Systems

System Base

Symbols Examples

Decimal

10 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

101, 33, 108, 987

Binary 2 0, 1 101, 1110, 10, 1, 0

Octal 8 0, 1, 2, 3, 4, 5, 6, 7

101, 33, 777, 642

Hexa-decima

l

16 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

A, B, C, D, E, F

101, 33, 108, 1AF

ABDF, 35FF, 10A

Conversion Among BasesConversion Among Bases


Divide the number by base and write the remainders.

Continue downwards, dividing each new quotient by base and writing the remainders.

Stop when the Stop when the quotient is 0.quotient is 0.


Start at the right Break the binary

numeral into groups of three digits

Replace each 3-digits binary numeral with it’s 1-digit octal equivalent

Replace each octal digit with it’s 3-digits binary equivalent

Combine all binary equivalent into a single binary numeral.

Octal Binary

Octal Binary


Start at the right Break the binary

numeral into groups of four digits

Replace each 4-digits binary numeral with it’s 1-digit hexadecimal equivalent

Replace each hexadecimal digit with it’s 4-digits binary equivalent

Combine all binary equivalent into a single binary numeral.

hexadecimal

Binary hexadecimal

Binary

Binary Addition and Binary Addition and MultiplicationMultiplication

Binary additionBinary addition

To add 3 or more numbers:To add 3 or more numbers:Add the first two numbers. Add the first two numbers. Then, add the third Then, add the third number to the result and so number to the result and so on.on.

ExamplesExamples

Binary Addition and Binary Addition and MultiplicationMultiplication

Binary MultiplicationBinary Multiplication

To multiply 3 or more To multiply 3 or more numbers:numbers:Multiply the first two Multiply the first two numbers. numbers. Then, multiply the result by Then, multiply the result by the third number and so on.the third number and so on.

ExampleExample

Chapter 2Chapter 2Digital Logic DesignDigital Logic Design

Digital Logic DesignDigital Logic Design

Logic gatesLogic gates Logic FunctionsLogic Functions Derivation of logical expressions Derivation of logical expressions

sum-of-products (SOP) formsum-of-products (SOP) form product-of-sums (POS) formproduct-of-sums (POS) form

Logical EquivalenceLogical Equivalence Truth table methodTruth table method Algebraic manipulation methodAlgebraic manipulation method

Logical Expression SimplificationLogical Expression Simplification Boolean AlgebraBoolean Algebra Karnaugh Map MethodKarnaugh Map Method

Logic gatesLogic gates

Logic gatesLogic gates AND OR NOT NANDNAND NORNOR XORXOR Precedence (NOT > AND > OR)Precedence (NOT > AND > OR)

You must know:You must know: The function and truth table of each gateThe function and truth table of each gate The graphical representation of each gateThe graphical representation of each gate The logical representation of each gateThe logical representation of each gate

Logic FunctionsLogic Functions Logical functions can be expressed in several ways:Logical functions can be expressed in several ways:

Truth tableTruth table Logical expressionsLogical expressions Graphical formGraphical form

You must know how to:You must know how to: Use a graphical representation to derive a logical expression.Use a graphical representation to derive a logical expression. Use a graphical representation to derive a truth table.Use a graphical representation to derive a truth table. Use a logical expression to derive a graphical representation.Use a logical expression to derive a graphical representation. Use a logical expression to derive a truth table.Use a logical expression to derive a truth table. Use a truth tables to derive a logical expression (SOP & POS)Use a truth tables to derive a logical expression (SOP & POS)

Derivation of logical Derivation of logical expressions expressions

An An SOP expression SOP expression when two or more when two or more product terms are product terms are summed by Boolean summed by Boolean addition.addition.

In an SOP form, a In an SOP form, a single overbar cannot single overbar cannot extend over more than extend over more than one variableone variable ExampleExample

But notBut not

An An POS expression POS expression When two or more sum When two or more sum terms are multiplied terms are multiplied by Boolean by Boolean multiplication.multiplication.

In a POS form, a single In a POS form, a single overbar cannot extend overbar cannot extend over more than one over more than one variable variable ExampleExample

But notBut not

DCABDCBACDBA

DCABDCBACDBA

))()(( DCBADCBADCBA

))()(( DCBADCBADCBA

Derivation of logical Derivation of logical expressions expressions

To determine the SOP expression represented by a truth table.Instructions:

Step 1: List the binary values of the input variables for which the output is 1.

Step 2: Convert each binary value to the corresponding product term by replacing:

each 1 with the corresponding variable, and

each 0 with the corresponding variable complement.

Example: 1010

To determine the POS expression represented by a truth table.Instructions:

Step 1: List the binary values of the input variables for which the output is 0.

Step 2: Convert each binary value to the corresponding product term by replacing:

each 1 with the corresponding variable complement, and

each 0 with the corresponding variable.

Example: 1001

DCBA DCBA

POSSOP

Derivation of logical Derivation of logical expressions from a Truth expressions from a Truth

Table (example)Table (example)I/PI/P O/PO/P

AA BB CC XX

00 00 00 00

00 00 11 00

00 11 00 00

00 11 11 11

11 00 00 11

11 00 11 00

11 11 00 11

11 11 11 11

There are There are four four 1s1s in the output in the output and the and the corresponding corresponding binary value binary value are 011, 100, are 011, 100, 110, and 111.110, and 111.

ABC

CAB

CBA

BCA

111

110

100

011

There are There are four four 0s0s in the output in the output and the and the corresponding corresponding binary value binary value are 000, 001, are 000, 001, 010, and 101.010, and 101.

CBA

CBA

CBA

CBA

101

010

001

000

ABCCABCBABCAX

))()()(( CBACBACBACBAX

Converting SOP and POS Converting SOP and POS Expressions to Truth Table Expressions to Truth Table

FormatFormat Recall the fact:Recall the fact:

An SOP An SOP expression expression corresponds to 1 output.corresponds to 1 output.

Constructing a truth table:Constructing a truth table: Step 1:Step 1: List all possible List all possible

combinations of binary combinations of binary values of the variables in values of the variables in the expression.the expression.

Step 2:Step 2: Place a 1 in the Place a 1 in the output column (X) for output column (X) for each binary value that each binary value that makes the makes the SOPSOP expression expression a 1 and place 0 for all the a 1 and place 0 for all the remaining binary values. remaining binary values.

Recall the fact:Recall the fact: A POS expression A POS expression

corresponds to 0 output.corresponds to 0 output. Constructing a truth table:Constructing a truth table:

Step 1:Step 1: List all possible List all possible combinations of binary combinations of binary values of the variables in values of the variables in the expression.the expression.

Step 2:Step 2: Place a 0 in the Place a 0 in the output column (X) for each output column (X) for each binary value that makes the binary value that makes the POSPOS expression a 0 and expression a 0 and place 1 for all the place 1 for all the remaining binary remaining binary values. values.

Converting SOP Converting SOP Expressions to Truth Table Expressions to Truth Table

Format (example)Format (example) Develop a truth Develop a truth

table for the table for the standard SOP standard SOP expressionexpression

ABCCBACBA

InputsInputs OutputOutput Product Product TermTermAA BB CC XX

00 00 00

00 00 11

00 11 00

00 11 11

11 00 00

11 00 11

11 11 00

11 11 11


00 00 00

00 00 11

00 11 00

00 11 11

11 00 00

11 00 11

11 11 00

11 11 11

CBA

CBA

ABC


00 00 00

00 00 11 11

00 11 00

00 11 11

11 00 00 11

11 00 11

11 11 00

11 11 11 11

CBA

CBA

ABC


00 00 00 00

00 00 11 11

00 11 00 00

00 11 11 00

11 00 00 11

11 00 11 00

11 11 00 00

11 11 11 11

CBA

CBA

ABC

Converting POS Converting POS Expressions to Truth Table Expressions to Truth Table

Format (example)Format (example) Develop a truth Develop a truth

table for the table for the standard SOP standard SOP expressionexpression

))((

))()((

CBACBA

CBACBACBA


00 00 00

00 00 11

00 11 00

00 11 11

11 00 00

11 00 11

11 11 00

11 11 11


00 00 00

00 00 11

00 11 00

00 11 11

11 00 00

11 00 11

11 11 00

11 11 11

)( CBA

)( CBA

)( CBA

)( CBA

)( CBA


00 00 00 00

00 00 11

00 11 00 00

00 11 11 00

11 00 00

11 00 11 00

11 11 00 00

11 11 11

)( CBA

)( CBA

)( CBA

)( CBA

)( CBA


00 00 00 00

00 00 11 11

00 11 00 00

00 11 11 00

11 00 00 11

11 00 11 00

11 11 00 00

11 11 11 11

)( CBA

)( CBA

)( CBA

)( CBA

)( CBA

Implementation of SOP Implementation of SOP & POS & POS

Implementation Implementation of an SOPof an SOP

Implementation Implementation of a POSof a POS

A

B

BA

A

B

X

A

B

B

A

A

B

ABBABA ))()(( BABABA

Logical EquivalenceLogical Equivalence

Truth table Truth table methodmethod Derive the logical Derive the logical

expressionexpression Derive truth tables Derive truth tables

for each expression. for each expression. If both expressions If both expressions

yield the same yield the same output, they are output, they are equivalent. equivalent. Otherwise, they are Otherwise, they are not.not.

Algebraic Algebraic manipulation manipulation methodmethod Derive the logical Derive the logical

expressionsexpressions Simplify each expression Simplify each expression

using boolean laws.using boolean laws. If both expressions yield If both expressions yield

the same simplified the same simplified expression, they are expression, they are equivalent. Otherwise, equivalent. Otherwise, they are not.they are not.

Logical Expression Logical Expression SimplificationSimplification

Boolean AlgebraBoolean Algebra

Need boolean identities Need boolean identities (Laws)(Laws)

Start with an Start with an expression and apply expression and apply Boolean laws to derive Boolean laws to derive the simplest (minimum) the simplest (minimum) expression possible.expression possible.

Karnaugh Map MethodKarnaugh Map Method

A K-map is a graphical A K-map is a graphical method for simplifying method for simplifying Boolean expressions Boolean expressions and, if properly used, and, if properly used, will produce the simplest will produce the simplest (minimum) expression (minimum) expression possible.possible.

The size of k-map The size of k-map depends on the number depends on the number of variables.of variables.

Simplification using Simplification using Boolean AlgebraBoolean Algebra

Simplification using Simplification using Boolean Algebra (cont.)Boolean Algebra (cont.)

Simplification using K-MapSimplification using K-Map

The process of simplification The process of simplification ((minimization):minimization): Mapping the expression into k-mapMapping the expression into k-map Grouping the 1sGrouping the 1s Determining the minimum SOP Determining the minimum SOP

expression from the mapexpression from the map

Grouping the 1s (rules)Grouping the 1s (rules)

1.1. A group must contain either 1,2,4, or 8 cells A group must contain either 1,2,4, or 8 cells (depending on number of variables in the (depending on number of variables in the expression)expression)

2.2. Each cell in a group must be adjacent to Each cell in a group must be adjacent to one or more cells in that same group.one or more cells in that same group.

3.3. Always include the largest possible number Always include the largest possible number of 1s in a group in accordance with rule 1.of 1s in a group in accordance with rule 1.

4.4. Each 1 on the map must be included in at Each 1 on the map must be included in at least one group. least one group.

5.5. The 1s already in a group can be included in The 1s already in a group can be included in another group as long as the overlapping another group as long as the overlapping groups include non common 1s.groups include non common 1s.

ABAB

CC0000 0101 1111 1010

00

11

Cell AdjacencyCell Adjacency

CC

ABAB00 11

0000

0101

1111

1010

Simplification using K-MapSimplification using K-Map(full example)(full example)

The expression: The expression:

CBACABCBACBA 000 001 110 100

1 1

1

1

DCBADCBADCABABCDDCABDCBACDBA

CBACBABCA

ABCCABCBACBA

Practice:

Simplification using K-MapSimplification using K-Map(full example)(full example)

CC

ABAB00 11

0000

0101

1111

1010 B

1 1

0 0

1 0

11

CAB

CA

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computer programming skills revision prepared by: ghader kurdi

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