Boolean Algebra

Instructor:Khaled Ibrahim

Binary Logic and GatesBinary variables:

take on one of two values.Logical operators:

operate on binary values and variablesLogic gates:

are symbolic representation for the logic functions.Boolean Algebra:

a useful mathematical system for specifying and transforming logical functions.

We study Boolean Algebra as foundation for designing digital systems.

Binary Variables

The two binary values have different names:

True/FalseOn/OffYes/No1/0

Variable identifiers:A, B, y, or z,…..

Logical Operations

The three basic logical operations are:ANDORNOT

AND is denoted by a dot (·)OR is denoted by a plus (+)NOT is denoted by a bar ( ¯ ) over the

variable

Notation Examples

Examples:Y = A⋅B is read “Y is equal to A and B.”z = x + y is read “z is equal to x OR y.”

is read “X is equal to NOT A.”AX =

Operator Definitions

Operations are defined on the values "0" and "1" for each Operator:

Operator Definitions

Using SwitchesFor inputs:

logic 1 is switch closedlogic 0 is switch open

For outputs:logic 1 is light onlogic 0 is light off.

Logic Gate Symbols

Truth TablesTruth tables list the output value of a function for all possible input valuesTruth tables for basic logic operations:

Logic Diagrams and Expressions

Boolean expressions, truth tables and logic diagrams describe the same function!Truth tables are unique, expressions and logic diagrams are not. This gives flexibility in implementing functions.

Boolean Algebra Rules

AA =+ 0

AA =⋅111=+A

00 =⋅A

Boolean Algebra Rules

AAA =+

AAA =⋅1=+ AA

0=⋅ AA

Boolean Algebra Rules

AA =

BABAA +=⋅+

ABAA =⋅+

CBACABA ⋅+=+⋅+ )()(

Boolean Algebra Rules

ABBA +=+

CBACBA ++=++ )()(

ABBA ⋅=⋅

CBACBA ⋅⋅=⋅⋅ )()(

Boolean Algebra Rules

BABA ⋅=+ )(

CABACBA ⋅+⋅=+⋅ )(

BABA +=⋅ )(

)()( CABACBA +⋅+=⋅+