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This chapter covers principles of sets and counting. After completing this chapter students should be able to: use set theory and venn diagrams to solve counting problems; use the multiplication axiom to solve counting problems; use permutations to solve counting problems; use combinations to solve counting problems; and use the binomial theorem to expand x+y^n.

Sets and counting

Find the indicated sets.

List all subsets of the following set.

Al, Bob size 12{ left lbrace "Al, Bob" right rbrace } {}
Al, Bob size 12{ left lbrace "Al, Bob" right rbrace } {} , Al size 12{ left lbrace "Al" right rbrace } {} , Bob , Ø size 12{ left lbrace "Bob" right rbrace } {}
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List all subsets of the following set.

Al, Bob, Chris size 12{ left lbrace "Al, Bob, Chris" right rbrace } {}
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List the elements of the following set.

Al, Bob, Chris, Dave Bob, Chris, Dave, Ed size 12{ left lbrace "Al, Bob, Chris, Dave" right rbrace intersection left lbrace "Bob, Chris, Dave, Ed" right rbrace } {}
Bob, Chris, Dave size 12{ left lbrace "Bob, Chris, Dave" right rbrace } {}
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List the elements of the following set.

Al, Bob, Chris, Dave Bob, Chris, Dave, Ed size 12{ left lbrace "Al, Bob, Chris, Dave" right rbrace union left lbrace "Bob, Chris, Dave, Ed" right rbrace } {}
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In Problems 5 - 8, let Universal set = U = a , b , c , d , e , f , g , h , i , j size 12{"Universal set"=U= left lbrace a,b,c,d,e,f,g,h,i,j right rbrace } {} , V = a , e , i , f , h size 12{V= left lbrace a,e,i,f,h right rbrace } {} , and W = a , c , e , g , i size 12{W= left lbrace a,c,e,g,i right rbrace } {} .

List the members of the following sets.

V W size 12{V union W} {}
a , e , i , f , h , c , g size 12{ left lbrace a,e,i,f,h,c,g right rbrace } {}
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V W size 12{V intersection W} {}
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V W ¯ size 12{ {overline {V union W}} } {}
b , d , j size 12{ left lbrace b,d,j right rbrace } {}
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V ˉ W ˉ size 12{ { bar {V}} intersection { bar {W}}} {}
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In 9 - 12, let Universal set = U = 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 size 12{"Universal set"=U= left lbrace a,b,c,d,e,f,g,h,i,j right rbrace } {} , A = 1,2,3,4,5 size 12{A= left lbrace 1,2,3,4,5 right rbrace } {} , B = 1,3,4,6 size 12{B= left lbrace 1,3,4,6 right rbrace } {} , and C = 2,4,6 size 12{C= left lbrace 2,4,6 right rbrace } {} .

List the members of the following sets.

A B size 12{A union B} {}
1,2,3,4,5,6 size 12{ left lbrace 1,2,3,4,5,6 right rbrace } {}
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A C size 12{A intersection C} {}
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A B ¯ C size 12{ {overline {A union B}} intersection C} {}

Ø

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A ˉ B C ¯ size 12{ { bar {A}} union {overline {B intersection C}} } {}
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Find the number of elements in the following sets.

In Mrs. Yamamoto's class of 35 students, 12 students are taking history, 18 are taking English, and 4 are taking both. Draw a Venn diagram and determine how many students are taking neither history nor English?

9 students

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In the County of Santa Clara 700,000 people read the San Jose Mercury News, 400,000 people read the San Francisco Examiner, and 100,000 read both newspapers. How many read either the Mercury News or the Examiner?

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A survey of athletes revealed that for their minor aches and pains, 30 used aspirin, 50 used ibuprofen, and 15 used both. How many athletes were surveyed?

65

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In a survey of computer users, it was found that 50 use HP printers, 30 use IBM printers, 20 use Apple printers, 13 use HP and IBM, 9 use HP and Apple, 7 use IBM and Apple, and 3 use all three. How many use at least one of these Brands?

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This quarter, a survey of 100 students at De Anza college finds that 50 take math, 40 take English, and 30 take history. Of these 15 take English and math, 10 take English and history, 10 take math and history, and 5 take all three subjects. Draw a Venn diagram and determine the following.

  1. The number of students taking math but not the other two subjects.
  2. The number of students taking English or math but not history.
  3. The number of students taking none of these subjects.
  1. 30
  2. 60
  3. 10
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In a survey of investors it was found that 100 invested in stocks, 60 in mutual funds, and 50 in bonds. Of these, 35 invested in stocks and mutual funds, 30 in mutual funds and bonds, 28 in stocks and bonds, and 20 in all three. Determine the following.

  1. The number of investors that participated in the survey.

  2. How many invested in stocks or mutual funds but not in bonds?

  3. How many invested in exactly one type of investment?

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Tree diagrams and the multiplication axiom

Do the following problems using a tree diagram or the multiplcation axiom.

A man has 3 shirts, and 2 pairs of pants. Use a tree diagram to determine the number of possible outfits.

6

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Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
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?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
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?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
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
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
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
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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8. It is known that 80% of the people wear seat belts, and 5% of the people quit smoking last year. If 4% of the people who wear seat belts quit smoking, are the events, wearing a seat belt and quitting smoking, independent?
William Reply
Mr. Shamir employs two part-time typists, Inna and Jim for his typing needs. Inna charges $10 an hour and can type 6 pages an hour, while Jim charges $12 an hour and can type 8 pages per hour. Each typist must be employed at least 8 hours per week to keep them on the payroll. If Mr. Shamir has at least 208 pages to be typed, how many hours per week should he employ each student to minimize his typing costs, and what will be the total cost?
Chine Reply
At De Anza College, 20% of the students take Finite Mathematics, 30% take Statistics and 10% take both. What percentage of the students take Finite Mathematics or Statistics?
Chalton Reply

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Source:  OpenStax, Applied finite mathematics. OpenStax CNX. Jul 16, 2011 Download for free at http://cnx.org/content/col10613/1.5
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