# 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 ?

#### Questions & Answers

what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
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
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
A fair die is tossed 180 times. Find the probability P that the face 6 will appear between 29 and 32 times inclusive