boolean algebra
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Boolean Algebra
Instructor:Khaled Ibrahim
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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.
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Binary Variables
The two binary values have different names:
True/FalseOn/OffYes/No1/0
Variable identifiers:A, B, y, or z,…..
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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
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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 =
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Operator Definitions
Operations are defined on the values "0" and "1" for each Operator:
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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.
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Logic Gate Symbols
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Truth TablesTruth tables list the output value of a function for all possible input valuesTruth tables for basic logic operations:
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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.
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Boolean Algebra Rules
AA =+ 0
AA =⋅111=+A
00 =⋅A
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Boolean Algebra Rules
AAA =+
AAA =⋅1=+ AA
0=⋅ AA
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Boolean Algebra Rules
AA =
BABAA +=⋅+
ABAA =⋅+
CBACABA ⋅+=+⋅+ )()(
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Boolean Algebra Rules
ABBA +=+
CBACBA ++=++ )()(
ABBA ⋅=⋅
CBACBA ⋅⋅=⋅⋅ )()(
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Boolean Algebra Rules
BABA ⋅=+ )(
CABACBA ⋅+⋅=+⋅ )(
BABA +=⋅ )(
)()( CABACBA +⋅+=⋅+