Contemporary Math 1029 Chpt 2

Empty (or Null) Set

A null (or empty set ) contains absolutely NO elements. These are examples of empty sets: - Set of all numbers less than 4 and greater than 10 - {x | x is a fawn that speaks}

Finite Set

A set that contains no elements or the number of elements in the set is a natural number. Example: Set S = {2, 3, 4, 5, 6, 7} is a finite set because the number of elements in the set is 6, and 6 is a natural number.

Chose the correct description of the set A = {4,8,12,16,....}

A= x|x e N and x is a multiple of 4}

Set A is the set of odd counting numbers smaller than 12. Set B is the set of odd natural numbers between 0 and 12.

A=B, A subset B, B subset A

List the elements in the set D U E, Let D (12,15,17) E=(12,14,15,16) and F =(11,13,14,15,17)

All elements in D and U 12,14,15,16,17

Determine whether the pair of sets is​ equal, equivalent,​ both, or neither. A-( computer science , history, geometry B- ( history, chemistry, computer science )

Are set A and set B equal, equivalent, both or neither? equivalent

Express the set in roster form B={x|xEN and x is a multiple of 7}

B={7,14,21,28,...}

Let U = {1,2,3,4,5,6,7} A = {1,5,6,7} and B= {1,3,7} Find the set (A U B)'

Combine A and B and then compare to U. List the letters that are not in A and B 2,4

Two sets with no elements in common are called​ ______ sets.

Disjoint

Express the given set in roster form. E={z|zeN and 18< x < 101}

E+{18,19,20,21,...100}

Express the following set in set-builder notation. E is the set of natural numbers that are multiples of 5

E={x|xeN and x is a multiple of 5}

Determine whether sets A and B are​ equal, equivalent,​ both, or neither. A is the set of letters in the word ​'flow​'. B is the set of letters in the word ​'comb​'.

Equivalent

Determine if the following statement is true. 8 = {}

False

Determine whether the set is finite or infinite. The set of odd numbers greater than 59

Infinite

Determine whether the set is finite or infinite. ​{28 comma 32 comma 36 comma 40 comma ...​}

Infinite- 28,32,36,40 is the roster notation of the set whose description is "all multiples of 4 greater than 27. If the number of elements is the set is a natural number, the set is finite. If the number of elements in the set cannot be counted, the set is infinite.

Determine whether the following statement is true or false. xe{k,p,m,v,b}

Is the statement true? No

Determine whether the pair of sets is​ equal, equivalent,​ both, or neither. AequalsStartSet calculus comma physics comma computer science EndSet BequalsStartSet physics comma literature comma computer science comma calculus EndSet Are set A and set B​ equal, equivalent,​ both, or​ neither?

Neither

Three Ways of Writing Set Example 3

Set-Builder notation: By using set-builder notation. {x | x is an even counting number less than 10} Set-builder Notation P = {x | x is the first five presidents of the United States}

Use the Venn diagram to determine the ( A' U B)

The letters in the conjoined circle and all the letters inside and outsife of B.

Use the roster method to write the set A' Let U = {5,6,7,8,9,10,11} Let A = {5,8,9,11}

This are the letters that are not duplicated in both sets. 6,7,10

Use the Venn diagram to determine the set U.

This is all of the letters inside the box.

Use the Venn diagram to list the set of elements in roster form. What are the elements of set (A U B)'

This is all the letters outside of both circles

Use the Venn diagram to list the set of elements in roster form What are the elements of set ( A u B)'?

This is the letters inside the circle A then out side the letter A then the letters that are in the overlapping circle.

Use the Venn diagram to determine the set (A n B)

This is the letters that are inside the B circle only

Decide if the given statement is true or false. If it is​ false, give the reason. ​{0}= {}

This statement is false. The set​ {0} contains the element 0.

Decide if the given statement is true or false. If it is​ false, give the reason. ​{kayak​, tug boat​} is a proper subset of ​{tug boat​, canoe​, kayak​, ferry​}

True

Decide if the given statement is true or false. If it is​ false, give the reason. ​{newspaper​} is a subset of ​{guidebook​, newspaper​, journal​}

True

Decide if the given statement is true or false. If it is​ false, give the reason. ​{orange​} is a proper subset of ​{orange​, purple​, indigo​}

True

Tell whether the statement is true or false. empty set equals { }

True

Decide if the given statement is true or false. If it is​ false, give the reason. Game Box is an element of ​{Advent​, Game Box​, Wizard​}

True GAme Box is not in {} so it is an element

A set that contains no elements is called the​ _______ set and is represented by​ _______.

empty/null & zero with a slash through it

Two sets that contain the same number of elements are called​ _______.

equivalent.

For the set A = {2,4,6} determine n(A)

n(A)=3

Cardinal Number

• The number of elements in set A is its cardinal number. • Symbol: n(A)

Universal Set

• The universal set contains all of the possible elements which could be under discussion in a particular problem. • Symbol: U

Let set A ={ h,q,x}. ​a) List all the subsets of set A.

​{}, ​{h​}, ​{q​}, ​{x​}, ​{h​, q​}, ​{h​, x​}, ​{q​, x​}, ​{h​, q​, x​}

List all the subsets of the given set. {lime,blueberry, lemon}

​{}, ​{lime​}, ​{blueberry​}, ​{lemon​}, ​{lime​, blueberry​}, ​{lime​, lemon​}, ​{blueberry​, lemon​}, ​{lime​, blueberry​, lemon​}

Infinite Set

• An infinite set contains an indefinite (uncountable) number of elements. • It is not possible to explicitly list out all the elements of an infinite set. • Whenever a set is so large that its cardinal number is not found among the whole numbers, we call that set an infinite set

Three Ways of Writing Set Example 1

1. Description method: By giving a verbal description of the set. The set of even counting numbers less than 10 Description Method P is the set of the first five presidents of the United States

Express the set in roster form The set of natural numbers between 15 and 199

{16,17,18,19,...,198}

Use roster form to write the elements of the following​ set, or state that the set has no elements. The zodiac signs that begin with Upper L

{Leo,Libra}

List all the subsets of the given set. empty set zero with line through it

{}

Three Ways of Writing Set Example 2

2. Roster form: Elements are all listed, each separated by commas inside a pair of braces, { }. {2, 4, 6, 8} Roster Form P = {Washington, Adams, Jefferson, Madison, Monroe}.

Determine whether A=​B, A is a subset of ​B, B is a subset of ​A, A is a proper subset of ​B, B is a proper subset of ​A, or if none of these applies.

B is a proper subset of A Your answer is correct. B is a subset of A

Express the set in roster form. The set of continents whose name begins with the letter Upper U

0 with a line through it

Write a description of the set. B = {1,2,3,4,5,6,7,8}

B is teh set of natural numbers less than or equal to 8

Write a description of the set. Bequals​{Leo comma Libra​}

B is the set of Zodiac signs that begin with L.

Aequals​{x ​| x is a sport that uses a ball​} Bequals​{tennis​, baseball​, racquetball​, volleyball​}

B proper subset of A B subset of A

Express the following set in​ set-builder notation. B={6, 7, 8, 9, 10, 11, 12,13}

B={x\xeN and 6< x <13} the < should have lines under then

Determine if the collection is not well defined and therefore is not a set. The collection of U.S. museums

Well defined

Determine whether the set is well defined. The set of U.S. cities

Well defined

What is a Set?

. A set is a collection of objects whose contents can be clearly determined.

Determining Cardinal Number

Example: Determine the cardinal number for the following sets A and B. A = {2, 3, 4, 5, 6, 7} B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} Solution: n(A) = 5 n(B)= 10

Tell whether the statement is true or false. {hockey comma soccer} subset equals ​{track​,hockey​,tennis​}

False

Determine if the following statement is true. ​{tomato​}is an element of​{beet​, lettuce​, tomato​}

False-elements are not in {}

Tell whether the statement is true or false. { 3 } is an element of{3. 4, 8}

False-elements are not in {}

Determine whether the following set is finite or infinite. A=The set of odd numbers between 0 and 13

Finite-because A is a smaller set than the set of all whole numbers between 0 and 13. Recall that a set is said to be finite if it either contains no elements or the number of elements in the set is a natural number. A set that is not finite is said to be infinite.

Which analysis applies to the statement {hockey comma soccer} subset equals {track comma hockey comma tennis}​?

Since soccer is not in the set {track, hockey, tennis}, {hockey, soccer} cannot be a subset of {track, hockey, tennis}

f all the elements of set A are also elements of set​ B, then set A is​ a(n) _______ of set B.

Subset-If all the elements of set A are also elements of set​ B, then set A is a subset of set B. The statement that set A is a subset of set B is symbolized by A is a subset of B. If it is also true that set Anot equalsset ​B, then set A is a proper subset of set B.

A set that contains all the elements for any specific discussion is called​ a(n) ________ set. The collection of the five worst presidents.

The collection is not well defined and therefore it is not a set.

Describe what is meant by the complement of a set.

The complement of a set A is the set of all elements in the universal set that are not in A. This is symbolized by Upper A prime.

The expression for determing the number of distinct subsetsfor a set with n distinct elements is 2n.

The expression for determining the number of distinct subsets for a set with n distinct elements is 2n. Note​ that, since every set is a subset of​ itself, but no set is a proper subset of​ itself, the number of distinct proper subsets for a set with n distinct elements is 2nminus1.

Use the Venn diagram to list the set of elements in roster form.A' U B'

The letters outside circle A, then letters inside the letter A circle, then the letters inside the circle B

Decide if the given statement is true or false. If it is​ false, give the reason. ​{9​, 7​, 4​} is a proper subset of ​{7​, 4​, 9​}

This statement is false. No set is a proper subset of itself.

Ellipsis

Three dots placed after a number to indicate that the elements in the set continue in the same manner up to and including the last element. For Example: • A = { 0, 2, 4, 6, ...} - Set A is set of even numbers • B = { 1, 2, 3, ..., 10} - Set B is set of first ten natural numbers

Use the Venn diagram to represent set A in roster form.

Use all numbers that are inside circle A even if circle A connects and crosses another circle.

A set that is not finite is called​ a(n) _________ set.

infinite

The set of elements common to both set A and set B is called the __________________ and sets A and B, and is symbolized by ___________________________

intersection and A upside down u B

Determine whether the following statement is true or false. {z}E​{q​, z​, v​, m​, c​}

is the statement true? No

Well-defined set

means that its contents can be clearly determined. No opinion is associated with the members. Example: Determine whether set A and B are well-defined sets. A = Set of Natural Numbers B = Set of Best Movies Solution: Set A is well-defined Set B is not a well-defined set as the opinion is associated with it.

A set that contains all the elements for any specific discussion is called​ a(n) ________ set.

universal

Equivalent Sets

wo finite sets A and B are said to be equivalent sets if cardinality of both sets are equal. i. e. n(A) = n(B) • Equivalent sets have the same number of elements in them. Written A ~ B or A ≡ B Note: Equivalent sets can be equal also. Equal sets have the exact same elements in them, regardless of their order.

Let set A ={ h,q,x}. State which of the subsets in part​ (a) are not proper subsets of set A.

{ h,q,x}.

List all the subsets of the given set. ​{chicken​, horse​}

{ }, ​{chicken​}, ​{horse​}, ​{chicken​, horse​}