Set Theory Symbols and Terminology Set – A collection of objects

Set TheorySymbols and Terminology

Set – A collection of objects

Element – An object in a set

Empty (Null) Set – A set that contains no elements

Cardinal Number (Cardinality) – The number of elements in a set

Finite Set – A set that contains a limited number of elements

Infinite Set – A set that contains an unlimited number of elements

There are three ways to describe a set

Word Description

Listing

Set Builder Notation

The following example shows the three ways we can describe the same set.

Word Description“The set of even counting numbers less

than 10”

ListingE = {2, 4, 6, 8 }

Set Builder NotationE = {x | x is an even counting number

that is less than 10 }

Cardinal Numbersn(E) means “the number of elements in

set E”

In this particular case n(E) = 4

Example 1) Suppose A is the set of all lower case letters of the alphabet. We could write out set A as follows:

A = {a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r s,t,u,v,w,x,y,z}

We can shorten this notation if we clearly show a pattern, as in the following:

A = {a,b,c,d,e…,w,x,y,z}

Try writing the following three sets by listing the elements.

1) The set of counting numbers between six and thirteen.

1)The set of counting numbers between six and thirteen.

B = {7, 8, 9, 10, 11, 12 }

2) C = {5, 6, 7, …, 13}

C = {5, 6, 7, 8, 9, 10, 11, 12, 13}

3) D = {x | x is a counting number between 6 and 7}

D = { } or Ø

Homework

• Page 54 # 1 - 10

Warm UpFind the cardinal number of each set

K = {2, 4, 8, 16}

M = { 0 }

R = {4, 5, …, 12, 13}

P = Ø

Warm UpFind the cardinal number of each set

K = {2, 4, 8, 16} n(K)=4

M = { 0 } n(M)=1

R = {4, 5, …, 12, 13} n(R)=10

P = Ø n(P)=0

Pg 54 #1-10 Answers

1. C 5. B

2. G 6. D

3. E 7. H

4. A 8. F

9. A = {1, 2, 3, 4, 5, 6}

10. B = {9, 10, 11, 12, 13, 14, 15, 16, 17}

Empty Set – Example)

P = {x | x is a positive number <0}

Therefore P = { } or P = Ø but

P ≠ {Ø}

Infinite Set – Example)

R = {y | y is an odd whole number}

Therefore R = {1, 3, 5, 7, …}

Finite Set – Example)

F = {z | z is a factor of 30}

Therefore F = {1, 2, 3, 5, 6, 10, 15, 30}

Classwork

& 55 #11 – 49 odd

11. The set of all whole numbers not greater than 4 can be expressed by listing as A ={0, 1,2,3, 4}.

13. In the set {6, 7,8.... , 14}, the ellipsis (three dots) indicates a continuation of the pattern. A complete listing oft his set is B ={6,7,8,9,10, 11, 12, 13,14}.

& 55 #11 – 49 odd15. The set { -15, -13, 11,..., -1} contains all

integers from -15 to -1 inclusive. Each member is two larger than its predecessor. A complete listing of this set is C ={- 15, -13, -11, -9, -7, -5, -3, -1}.

17. The set {2, 4, 8, ... , 256} contains all powers of two from 2 to 256 inclusive. A complete listing of this set

D={2, 4,8,16,32,64,128, 256}.

& 55 #11 – 49 odd19. A complete listing of the set {x x is an even

whole number less than 11 } is

E={0, 2, 4, 6, 8, 10}. Remember that 0 is the first whole number.

21. The set of all counting numbers greater than 20 is represented by the listing F={21, 22, 23,... }.

& 55 #11 – 49 odd23. The set of Great Lakes is represented by

G={Lake Erie, Lake Huron, Lake Michigan, Lake Ontario, Lake Superior}.

25. The set {x | x is a positive multiple of 5} is represented by the listing H={5, 10,15,20,.,. }.

& 55 #11 – 49 odd

27. The set {x|x is the reciprocal of a natural number} is represented by the listing

I={1, 1/2, 1/3, 1/4, 1/5, ... }.

29. The set of all rational numbers may be represented using set-builder notation as

J={x|x is a rational number}.

& 55 #11 – 49 odd31. The set {1, 3,5,... , 75} may be represented

using set- builder notation as K={x|x is an odd natural number less than 76}.

33. The set {2, 4, 6,... , 32} is finite since the cardinal number associated with this set is a whole number.

& 55 #11 – 49 odd35. The set {112, 2/3, 3/4, ... } is infinite since there

is no last element, and we would be unable to count all of the elements.

37. The set {x|x is a natural number greater than 50} is infinite since there is no last element, and therefore its cardinal number is not a whole number.

& 55 #11 – 49 odd39. The set {x|x is a rational number} is infinite

since there is no last element, and therefore its cardinal number is not a whole number;

41. For any set A, n(A) represents the cardinal number of the set, that is, the number of elements in the set. The set A = {0, 1, 2, 3, 4, 5, 6, 7} contains 8 elements. Thus, n(A) = 8.

& 55 #11 – 49 odd

43. The set A = {2, 4, 6, ... , l000} contains 500 elements. Thus, n(A) = 500.

45. The set A = {a, b, c, ,.. , z} has 26 elements (letters of the alphabet). Thus

n(A) = 26.

& 55 #11 – 49 odd

47. The set A = the set of integers between -20 and 20 has 39 members. The set can be indicated as {- 19, -18,...,18, 19}, or 19 negative integers, 19 positive integers, and 0. Thus, n(A) = 39.

49. The set A = { 1/3, 2/4, 3/5, 4/6, ..., 27/29, 28/30} has 28 elements. Thus, n(A) = 28.

Equal and Equivalent Sets

Equal Sets – Two sets are equal if they contain the EXACT same elements.

A={1,4,9,16,25}

B={1,9,4,25,16}

Equal and Equivalent Sets

Equivalent Sets – Two sets are equivalent if they contain the same NUMBER of elements.

A={1,3,5,7,9}

B={1,2,4,8,16}

Well Defined and Not Well Defined Sets

Not We

Well D

ll Defined

{ | is a pretty number}

efined

{ | is a prime number}A x x

Not Well D

Well Defin

{ | is a st

{ | is a good student in this cla

udent in this class}

ss}B x x

Defined

efined

| is a positi

{ | is a positive pe

ve number

rC x x

On your own, come up with one example of a well defined set and one example of a not well defined set. Place your sets in the appropriate section of the board.

Elements of Sets

The symbol means that the object in question is

an element of a particular set.

The symbol means that the object in question is

NOT an element of a particular set.

Elements of SetsFor example:

Since sets can not be ELEMENTS of

{ , , , ,

{ , , , , }

other sets

} { , , , , , }

a m i x e r

a w a s h

s h e r

Homework

• Do page 55 #53 – 84

• QUIZ tomorrow on pages 54-55 #1 - 84

Homework Answers

53. Well Defined

54. Well Defined

55. Not Well Defined

58. Well Defined

Tuesday Oct 5

• Quiz Today

• After Quiz, do page 56 Question #92. Finish it for homework and be prepared to turn it in.

Wednesday Oct 6Venn Diagrams and Subsets

Consider the set of counting numbers less than or equal to 20.

U={x|x is a counting number less than 20}

U={1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}

Use the following Venn Diagram to divide the numbers into groups of even numbers and groups of multiples of three.

Even Numbers Multiples of 3

UCounting Numbers Counting Numbers ≤ 20≤ 20

1416 18

Universal Set – The set of all objects under discussion. For our example, the universal set is the set of all counting numbers less than or equal to 20. The universal set is always denoted by the letter U

Let’s let A represent the set of all even numbers less than or equal to 20 and B will represent the set of all multiples of 3 that are less than or equal to 20

A = {2,4,6,8,10,12,14,16,18,20}

B = {3,6,9,12,15,18}

1416 18

Complement of a Set – The complement of a set is the set of all elements of the universal set that are NOT elements of the set in question. In our example the complement of A, written A´, is

A´={1, 3, 5, 7, 9, 11, 13, 15, 17, 19}

121416 18

Subset of a Set – The subset of a set is the set where ALL elements of one set are also elements another set.

Using our example, A is a subset of U. B is also a subset of U

U = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20}

A = {2,4,6,8,10,12,14,16,18,20}

U = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20}

B = {3,6,9,12,15,18}

means "A is a subset of U"

Is ? Why or why not?

means "A is a subset of U"

Is ? Why or why no

es, because every element of B

is also an element of U

Is ? Is ? Why or why not?A B B A

No because NOT EVERY element of A

is an element of B

No because NOT EVER

Is ? Is ?

Y element o

ement of A

?A B B A

1416 18

Example: Consider the following

{ , , , , ,... , , }

{ , , , , , }

{ , , }

U a b c d e x y z

D s a w y e r

F y e s

Wednesday Oct 6Venn Diagrams and SubsetsExample: Consider the following

{ , , , , ,... , , }

{ , , , , , }

No, because , and F

Yes, bec

Is ? ause , and D

U a b c d e x y z

D s a w y e r

F y e s

y e sF D

Wednesday Oct 6Homework

#1 – 14

Subsets and Proper SubsetsList all of the subsets

{ } or Ø

{a, b}

{a, b, c}

{a, b, c }

{a}; {b}; {c}

{a, b}; {a, c}; {b, c}

{a, b, c }

{a, b, c, d}

{a, b, c, d }

{a}; {b}; {c}; {d}

{a, b}; {a, c}; {a, d}; {b, c}; {b, d}; {c, d}

{a, b, c }; {a, b, d }; {a, c, d }; {b, c, d }

{a, b, c, d }

Subsets and Proper SubsetsList all of the proper subsets

{ } or Ø

{a, b}

{a, b, c}

{a, b, c }

{a}; {b}; {c}

{a, b}; {a, c}; {b, c}

{a, b, c, d}

{a, b, c, d }

{a}; {b}; {c}; {d}

{a, b}; {a, c}; {a, d}; {b, c}; {b, d}; {c, d}

{a, b, c }; {a, b, d }; {a, c, d }; {b, c, d }

Subsets and Proper Subsets

a set that contains n elements will have

(2 1) prop

2 subsets

er subsets

Homework Page 61-62 #15-53

False 35.

True 33.

True 31.

True 29.

True 27.

False 25.

True 23.

neither 21.

both 19.

both 15.

10} 9, 8, 7, 6, 5, 4, 3, 2, {1, U53.

{2} 51.

10} 9, 7, 5, 3, {2, 49.

subsetsproper 31

subsets 32 47.

subsetsproper 63

subsets 64 45.

subsetsproper 7

subsets 8 43.

False 41.

True .39

False 37.

#55 - 60

#55 - 60U = {Higher Cost, Lower Cost, Educational, More Time to See Sights, Less Time to See Sights, Cannot Visit Relatives, Can Visit Relatives}

F = {Higher Cost, Educational, More Time to See Sights, Cannot Visit Relatives}

F΄ = {Lower Cost, Less Time to See Sights, Can Visit Relatives}

#55 - 60U = {Higher Cost, Lower Cost, Educational, More Time to See Sights, Less Time to See Sights, Cannot Visit Relatives, Can Visit Relatives}

D = {Lower Cost, Educational, Less Time to See Sights, Can Visit Relatives}

D΄ = {Higher Cost, More Time to See Sights, Cannot Visit Relatives}

#55 - 60F = {Higher Cost, Educational, More Time to See Sights, Cannot Visit Relatives}

D = {Lower Cost, Educational, Less Time to See Sights, Can Visit Relatives}

Both F and D = {Educational}

#55 - 60F΄ = {Lower Cost, Less Time to See Sights, Can Visit Relatives}

Both F΄ and D ΄ = Ø

#55 - 60F = {Higher Cost, Educational, More Time to See Sights, Cannot Visit Relatives}

Both F and D ΄ = {Higher Cost, More Time to See Sights, Cannot Visit Relatives}

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