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Shankar Balachandran* Associate Professor, CSE Department Indian Institute of Technology Madras
*Currently a Visiting Professor at IIT Bombay
x 1 = x = (a) Two states of a switch S x (b) Symbol for a switch
Digital Circuits and Systems Spring 2015 Week 1 Module 2 Shankar - - PowerPoint PPT Presentation
Digital Circuits and Systems Spring 2015 Week 1 Module 2 Shankar - - PowerPoint PPT Presentation
Digital Circuits and Systems Spring 2015 Week 1 Module 2 Shankar Balachandran* Associate Professor, CSE Department Indian Institute of Technology Madras *Currently a Visiting Professor at IIT Bombay Binary Switch x = 0 x = 1 (a) Two
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Simple connection to a battery S Battery Light
x
A Light Controlled by a Switch
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(a) The logical AND function (series connection) S Power supply S S Power supply S (b) The logical OR function (parallel connection) Light Light x1 x2 x1 x2
Two Basic Functions
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S Power supply S Light S
x1 x2 x3
A Series Parallel Circuit
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S Light Power supply R
x
An Inverting Circuit
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Truth Table
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Truth Table of 3-Input AND and OR Operations
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Truth Table of 3-Input AND and OR Operations
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Introduction 13
Symbols
AND Dot ( ) Imagine it to be like multiplication Example x y Called “x and y” OR Plus ( + ) Imagine it to be like addition Example x + y Called “x or y”
v
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NOT Operation
Symbol Closing single quote ’ Also overline
and ! symbol
Example: x’, x, !x Calling x complement “not of x” Simpler: “x bar”
Introduction 14
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Symbols
Introduction 15
x
1
x
2
x
n
x
1 x 2
x
n
+ + + x
1
x
2
x
1 x 2
+ x
1
x
2
x
n
x
1
x
2
x
1 x 2
x
1 x 2 x n
(a) AND gates (b) OR gates
x x
(c) NOT gate
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Boolean Algebra
Named after George Boole Axioms 0 · 0 = 0 0 + 0 = 0 0’ = 1
Duality
1 · 1 = 1 1 + 1 = 1 1’ = 0
0 + 1 = 1 + 0 = 1 0 · 1 = 1 · 0 = 0
Introduction 16
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Single Variable Theorems
x · 0 = 0 x + 1 = 1 x · 1 = x x + 0 = x x · x = x x + x = x x + !x = 1 x · !x = 0 x · x · x · … x = x !!x = x
Introduction 17
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Two Variable Theorems
x · y = y · x x + y = y + x Both are commutative
Introduction 18
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Three Variable Theorems
Associative Laws
x · (y · z) = (x · y) · z
x + (y + z) = (x + y) + z
Distributive Law
x · (y + z) = (x · y) + (x · z)
x + (y · z) = (x + y) · (x + z) More as we go
Introduction 19
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End of Week 1: Module 2
Thank You
Introduction 20