Boolean Algebra and Logic Gates-Digital Logic Design-Lecture Slides, Slides of Digital Logic Design and Programming

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Chapter 2 of Morris Mano

Binary logic uses two possible values
‘1’ and ‘0’

‘yes’ and ‘no’‘true’ and ‘false’
Can be represented by variables: A, B, C, x, y, z,… Logic functions modify input values

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Logic functions modify input values
What is the minimum number of inputs?

1 input -> pretty uninteresting, inverter or wire only2 inputs -> basic logic functions
What are possible logic functions?
NOT, AND, OR
Others can be derived from those (NAND, XOR,etc.)

Operation to check if two conditions aresatisfied

Two inputsOutput = 1 if and only if both inputs are 1

Symbolized by dot or absence of operator x·y or xy

x y z

inputs output

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x·y or xy

Truth table
Needs to consider 2 2 = 4 input combinations

Graphic symbol
AND gate

x y z

Operation to check if at least one conditionis met

Two inputsOutput = 1 if any one or both inputs are 1

Symbolized by “plus” sign: x + y x y z

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Truth table

Graphic symbol
OR gate

Two inputs might not be enough3-input AND gate:

4-input OR gate:

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4 - input OR gate:

What are the rules for aggregating functions?
Boolean algebra

Basic Gates
AND

ORNot

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Extended Gates

NANDNOR

XORXNOR

Operation to check if twoconditions are not met (for two input)
Output = 1 if and only if both inputs
are 0Symbolized by + operator with a^00

Inputs x y Output z

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Symbolized by + operator with a bar over them x+y
Truth table
Graphic symbol
NAND gate

01 01 1 1

(^00) 0

Operation to check if two conditions arenot met (for two input)

■ ■ Output = 1 if one of the inputs is 1Symbolized by operator

Truth table^ x^ y^ Inputs^ x^ y^^ Output z

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Truth table

Graphic symbol ■ XOR gate

xy z

x y z

Equation F =X +Y Z

Truth Table

0 1 00 0 1 01

X Y Z0 0 0 F^ = X 0 + Y ×Z

X Y F Z

Logic Diagram 1 1 01 1 1 11

1 0 01 0 1 11

0 1 00 1 1 00

Boolean equations, truth tables and logic

diagrams describe the same function!

Truth tables are unique; expressions and

logic diagrams are not. This gives flexibility

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logic diagrams are not. This gives flexibility

in implementing functions.

(E.g. NAND gate, NOT AND gate are same

their truth tables are unique but

expressions and logic diagrams are not)

Father of Boolean algebraHe came up with a type of linguistic algebra, the three most basic operations of which were (andstill are) AND, OR and NOT. It was these three functions that formed the basis of his premise,and were the only operations necessary to perform comparisons or basic mathematicalfunctions.
Boole’s system (detailed in his 'An Investigation of

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Boole’s system (detailed in his 'An Investigation of the Laws of Thought, on Which Are Founded the Mathematical Theories of Logic and Probabilities',1854) was based on a binary approach, processing only two objects - the yes-no, true-false, on-off, zero-one approach.
Surprisingly, given his standing in the academiccommunity, Boole's idea was either criticized or
completely ignored by the majority of his peers.Eventually, one bright student, Claude Shannon (1916-2001), picked up the idea and ran with it

George Boole (1815 - 1864)

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