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A fundamental problem is to determine the probability of a logical (Boolean) combination of a finite class of events, when the probabilities of certain other combinations are known. If we partition an event F into component events whose probabilities can be determined, then the additivity property implies the probability of F is the sum of these component probabilities. Frequently, the event F is a Boolean combination of members of a finite class -- say {A, B, C} or {A, B, C,D}. For each such finite class, there is a fundamental partition determined by the class. The members of this partition are called minterms. Any Boolean combination of members of the class can be expressed as the disjoint union of a unique subclass of the minterms. If the probability of every minterm in this subclass can be determined, then by additivity the probability of the Boolean combination is determined. An important geometric aid to analysis is the minterm map, which has spaces for minterms in an orderly arrangement.

Introduction

A fundamental problem in elementary probability is to find the probability of a logical (Boolean) combination of a finite class of events, when the probabilities ofcertain other combinations are known. If we partition an event F into component events whose probabilities can be determined, then the additivity property implies the probability of F is the sum of these component probabilities. Frequently, the event F is a Boolean combination of members of a finite class– say, { A , B , C } or { A , B , C , D } . For each such finite class, there is a fundamental partition determined by the class. The members of this partition are called minterms . Any Boolean combination of members of the class can be expressed as the disjoint union of a unique subclass of the minterms. If the probability of every mintermin this subclass can be determined, then by additivity the probability of the Boolean combination is determined. We examine these ideas in more detail.

Partitions and minterms

To see how the fundamental partition arises naturally, consider first the partition of the basic space produced by a single event A .

Ω = A A c

Now if B is a second event, then

A = A B A B c and A c = A c B A c B c , so that Ω = A c B c A c B A B c A B

The pair { A , B } has partitioned Ω into { A c B c , A c B , A B c , A B } . Continuation is this way leads systematically to a partition by three events { A , B , C } , four events { A , B , C , D } , etc.

We illustrate the fundamental patterns in the case of four events { A , B , C , D } . We form the minterms as intersections of members of the class, with various patterns of complementation.For a class of four events, there are 2 4 = 16 such patterns, hence 16 minterms. These are, in a systematic arrangement,

A c B c C c D c A c B C c D c A B c C c D c A B C c D c
A c B c C c D A c B C c D A B c C c D A B C c D
A c B c C D c A c B C D c A B c C D c A B C D c
A c B c C D A c B C D A B c C D A B C D

No element can be in more than one minterm, because each differs from the others by complementation of at least one member event. Eachelement ω is assigned to exactly one of the minterms by determining the answers to four questions:

Is it in A ? Is it in B ? Is it in C ? Is it in D ?

Questions & Answers

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Maciej
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s. Reply
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Devang Reply
are you nano engineer ?
s.
what is the Synthesis, properties,and applications of carbon nano chemistry
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Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
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s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
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s.
Graphene has a hexagonal structure
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Cied
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I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
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Yasmin
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Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
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what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
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AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
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after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
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silver nanoparticles could handle the job?
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Azam
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Uday
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Source:  OpenStax, Applied probability. OpenStax CNX. Aug 31, 2009 Download for free at http://cnx.org/content/col10708/1.6
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