# Minterms

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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, $\left\{A,\phantom{\rule{0.166667em}{0ex}}B,\phantom{\rule{0.166667em}{0ex}}C\right\}$ or $\left\{A,\phantom{\rule{0.166667em}{0ex}}B,\phantom{\rule{0.166667em}{0ex}}C,\phantom{\rule{0.166667em}{0ex}}D\right\}$ . 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 .

$\Omega =A\bigvee {A}^{c}$

Now if B is a second event, then

$A=AB\bigvee A{B}^{c}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{and}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{A}^{c}={A}^{c}B\bigvee {A}^{c}{B}^{c},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{so}\phantom{\rule{4.pt}{0ex}}\text{that}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\Omega ={A}^{c}{B}^{c}\bigvee {A}^{c}B\bigvee A{B}^{c}\bigvee AB$

The pair $\left\{A,\phantom{\rule{0.166667em}{0ex}}B\right\}$ has partitioned Ω into $\left\{{A}^{c}{B}^{c},\phantom{\rule{0.166667em}{0ex}}{A}^{c}B,\phantom{\rule{0.166667em}{0ex}}A{B}^{c},\phantom{\rule{0.166667em}{0ex}}AB\right\}$ . Continuation is this way leads systematically to a partition by three events $\left\{A,\phantom{\rule{0.166667em}{0ex}}B,\phantom{\rule{0.166667em}{0ex}}C\right\}$ , four events $\left\{A,\phantom{\rule{0.166667em}{0ex}}B,\phantom{\rule{0.166667em}{0ex}}C,\phantom{\rule{0.166667em}{0ex}}D\right\}$ , etc.

We illustrate the fundamental patterns in the case of four events $\left\{A,\phantom{\rule{0.166667em}{0ex}}B,\phantom{\rule{0.166667em}{0ex}}C,\phantom{\rule{0.166667em}{0ex}}D\right\}$ . 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,

 ${A}^{c}{B}^{c}{C}^{c}{D}^{c}$ ${A}^{c}B{C}^{c}{D}^{c}$ $A{B}^{c}{C}^{c}{D}^{c}$ $AB{C}^{c}{D}^{c}$ ${A}^{c}{B}^{c}{C}^{c}D$ ${A}^{c}B{C}^{c}D$ $A{B}^{c}{C}^{c}D$ $AB{C}^{c}D$ ${A}^{c}{B}^{c}C\phantom{\rule{0.277778em}{0ex}}{D}^{c}$ ${A}^{c}BC\phantom{\rule{0.277778em}{0ex}}{D}^{c}$ $A{B}^{c}C\phantom{\rule{0.277778em}{0ex}}{D}^{c}$ $ABC\phantom{\rule{0.277778em}{0ex}}{D}^{c}$ ${A}^{c}{B}^{c}C\phantom{\rule{0.277778em}{0ex}}D$ ${A}^{c}BC\phantom{\rule{0.277778em}{0ex}}D$ $A{B}^{c}C\phantom{\rule{0.277778em}{0ex}}D$ $ABC\phantom{\rule{0.277778em}{0ex}}D$

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 ?

where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
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A fair die is tossed 180 times. Find the probability P that the face 6 will appear between 29 and 32 times inclusive