MATH 104 Chapter 2

Sets

Notation and overview2.1 BasicsǾ Empty set

∈ Is an elementIs not an element of

2.2 Subsets Is a subset of

Is not a subset of Is a proper subset ofË Is not a proper subset of

2.3 Set Operations Intersection Union

Ways to represent a set:

Roster method: V={a,e,i,o,u} A={a,b,c,d,e,...x,y,z} D={Sun,Mon,Tues,...Fri,Sat} N={1,2,3,4,...} natural numbers Set builder notation: D={x| } Empty set { } or Ǿ Notation

Definitions

The cardinal number is the number of elements in a set.

Finite setthe number of elements is a natural number or 0

Exercises from 2.1

1. True or False from section 2.1 a. a {a,e,I,o,u}∈b. 4 {x| x N } ∈ ∈c. 4 {x| x N and x is odd}∈ ∈

2. Find the cardinal number of each.a. V={x | x is a vowel} n(V)=b. A=(x |x is a letter of the alphabet} n(A)=

3. Are the following sets finite or infinite?a. V={x | x is a vowel}b. A=(x |x is a letter of the alphabet}c. N={x | x is a natural number}

Section 2.2- Subsets

Examples: Let B={x|x is a resident of PA}Find some subsets of B Def: Set A is a subset of set B, A B, if every

element in set A is also an element in set B.

Def: Set A is a proper subset of set B, A B, if A is smaller than B and if every element in set A is also an element in set B.

1. Element and subset notation from section 2.2: Are the following true or false?

a. Pennsylvania {x | x is one of the United State}b. Pennsylvania {x | x is one of the United States}c. {Pennsylvania} {x | x is one of the United States}d. {Pennsylvania} {x | x is one of the United States}e. {Pennsylvania, Ohio} {x | x is one of the United States}f. {Pennsylvania, Ohio} {x | x is one of the United States}g. { } = ∅h. { } = ∅ ∅i. a {a,b,c,}∈h. {a,b} {a,b,c,}i. {a,b} {a,b,c,}j. {a,b,c} {a,b,c,}k. {a,b,c} {a,b,c,}l. { } {a,b,c,}∈m. { } {a,b,c,}n. { } {a,b,c,}

Find all subsets of {a,b}

2. Find all subsets of {a,b} 3. Find all proper subsets of {a,b} (Proper subsets of B

are subsets of B that are not equal to B. That is, they are smaller)

4. Find all subsets of {a,b,c}

5. How many subsets would {a,b,c,d} have?

• Do you see a pattern?

Section 2.3- Set Operations:

Def: A' = complement of A A' = intersection: A B union: A B

Set Operations:

1. For the following diagram, answer the questionsAll students in a tutoring group: U={Anne, Bob, Cindy, Dave, Ellen}Students who studied Saturday: A={Anne, Bob, Cindy}Students who studied Sunday: B={Cindy, Dave}List and describe the students who are in: A B =A B =A’ =B’ =(A B)’ =(A B)’ =

2. For the following, answer…U={1,2,3,4,5,6,7,8,9,10},A={1,3,5,7,9},B={1,2,3,4,5}

A B= A B=A’ = B ‘ = (A B) ‘ = A’ B ‘ = (A B) ‘ = A’ B ‘ =

Sec 2.4-- Set ops - 3 sets; Sec 2.5--Surveys

1. Define U =all students in

this class, A = all females, B =all freshmen, andC =Porreco students.

Draw 3 circles and place yourself.

..

2. Define U =all students in

this class, A =blonds, B =criminal justice

majors, and C = fulltime

students. Draw 3 circles and place yourself.

Locate the elements

A

C B

Let U={a,b,c,d,e,f,g,h,i,j,k}A={a,b,d,g,k}B={b,c,d,f,i}C={b,h,k,i}Find:A BB CA CA B C

…more examplesLet U={a,b,c,d,e,f,g,h,i,j,k}A={a,b,d,g,k},B={b,c,d,f,i}C={b,h,k,i}. Find:A ‘ B ‘ A B(A B ) ‘A B(A B ) ‘A ‘ B ‘A ‘ B ‘

A

C B

Page 80 – example from the text:

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Page 84 in the text. Find drawing in the text. Answer #33-#44

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Venn1. (#44.) A survey of 120 college

students was taken at registration. Of those surveyed, 75 students registered for a math course, 65 for an English course, and 40 for both math and English. Of those surveyed,

a. How many registered only for a math course?

b. How many registered only for an English course?

c. How many registered for a math course or an English course?

d. How many did not register for either a math course of an English course?

#22. (#46) A survey of 180 college men was

taken to determine participation in various campus activities. 43 were in fraternities, 52 in sports, 35 in tutorial programs. 13 were in fraternities and sports, 14 in sports and tutorial, 12 in fraternities and tutorial, and 5 were in all three. Of those surveyed, how many participated in:

a. Only sports?b. Fraternities and sports, but not

tutorial?c. Fraternities or sports, but not tutorial?d. Exactly one of these activities?e. At least two of these activities?f. None of these activities?

#3

3. (#60) A person applying for the position of college registrar submitted the following report on 90 students: 31 take math, 28 take chemistry, 42 take psychology, 9 take math and chemistry, 10 take chemistry and psychology, 6 take math and psychology, 4 take all three subject, and 20 taken none of the courses. The applicant was not hired. Explain why.

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