introductions to sets

8/2/2019 Introductions to Sets

1/36

Introductions toSets

Unit 15>Lesson 1 of 14

Example 1: Kyesha was in math class with her friend Angie. Shewhispered to Angie that she had just bought a set ofwinter clothes. The outerwear collection includes a coat,a hat, a scarf, gloves, and boots. Their teacher, Mrs.Glosser, overheard the conversation and askedthem: What is a set?

Solution: Luckily for Kyesha and Angie, their classmate Eduardohad a math dictionary with him! He quickly looked up theword "set" and defined it for the class as shown below.

A set is a collection of objects that have something in common or follow a rule. The objects inthe set are called its elements. Set notation uses curly braces, with elements separated bycommas. So the set of outwear for Kyesha would be listed as follows:

A = {coat, hat, scarf, gloves, boots}, where A is the name of the set, and the braces indicatethat the objects written between them belong to the set.

Every object in a set is unique: The same object cannot be included in the set more thanonce.

Let's look at some more examples of sets.

Example 2: What is the set of all fingers?

Solution: P= {thumb, index, middle, ring, little}

Note that there are others names for these fingers: Theindex finger is commonly referred to as the pointer finger;the ring finger is also known as the fourth finger, and thelittle finger is often referred to as the pinky. Thus, wecould have listed the set of fingers as:

P= {thumb, pointer, middle, fourth, pinky}

Example 3: What is the set of all even whole numbersbetween 0 and 10?

Solution: Q= {2, 4, 6, 8} Note that the use of theword betweenmeans that the range of
http://www.mathgoodies.com/lessons/toc_unit15.htmlhttp://www.mathgoodies.com/lessons/toc_unit15.htmlhttp://www.mathgoodies.com/lessons/toc_unit15.html


2/36

numbers given is notinclusive. As a result, thenumbers 0 and 10 are not listed as elementsin this set.

Example 4: Eduardo was in art class when the teacher wrote thison the chalkboard: In fine arts, primary colors are setsof colors that can be combined to make a usefulrange of colors. Then she asked the class: What isthe set of primary colors?

Solution: Eduardo answered: red, blue and yellow. Angieanswered: We can use set notation to list the set of allprimary colors. Kyesha went to the chalkboard andwrote:

X= {red, blue, yellow}

The teacher said: Good work everyone. This is a nicecombination of art and math!

In examples 1 through 4, each set had a different number of elements, and each element withina set was unique. In these examples, certain conventions were used.

The following conventions are used with sets:

Capital letters are used to denote sets.

Lowercase letters are used to denote elements of sets. Curly braces { } denote a list of elements in a set.

So for examples 1 through 4, we listed the sets as follows:

1. A = {coat, hat, scarf, gloves, boots}2. P= {thumb, index, middle, ring, little}3. Q= {2, 4, 6, 8}4. X= {red, blue, yellow}

These sets have been listed with roster notation. Roster notation is a list of elements,separated by commas, enclosed in curly braces. The curly braces are used to indicate that theelements written between them belong to that set. Let's look at some more examples of setslisted with roster notation.

Example 5: Let Rbe the set of all vowels in the English alphabet.
http://popupwindow%28%27inclusive%27%29/http://popupwindow%28%27inclusive%27%29/http://popupwindow%28%27inclusive%27%29/http://popupwindow%28%27inclusive%27%29/


3/36

Solution: R= {a, e, i, o, u}

Example 6: Let Gbe the set of all whole numbers less than ten.

Solution: G= {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}

Example 7: Let Tbe the set of all days in a week.

Solution: T= {Monday, Tuesday, Wednesday, Thursday, Friday}

Example 8: Let Xbe the set of odd numbers less than 12.

Solution: X= {1, 3, 5, 7, 9, 11}

Example 9: Let Ybe the set of all continents of the world.

Solution: Y= {Asia, Africa, North America, South America, Antarctica, Europe, Australia}

There are times when it is not practical to list all the elements of a set. In this case, it is betterto describe the set. The rule that the elements follow can be given in the braces. Forexample,:

R= {vowels} means Let R be the set of all vowels in the English alphabet.

This is especially useful when working with large sets, as shown below.

A = {types of triangles}

G= {letters in the English alphabet}

J= {prime numbers less than 100}

M= {state capitals in the US}

When describing a set, It is not necessary to list every element in that set. Thus, there are two

methods for indicating a set of objects: 1) listing the elements and 2) describing the elements.We will distinguish between these two methods in examples 10 and 11 below.

Example 10: What is the set of all letters in the English alphabet?

Listing elements: D= {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}

Describing elements: D= {letters in the English alphabet}


4/36

Example 11: What is the set of all states in the Unites States?

Solution: R= {all states in the US}

In example 10, set Dhas 26 elements, so it is easier to describe its elements than to list them.Similarly, in example 11, setRhas 50 elements, so it is easier to describe its elements.

Summary: A set is a collection of objects that have something in common or follow a rule.The objects in the set are called its elements. Curly braces are used to indicatethat the objects written between them belong to a set. Every object in a set isunique. It is not necessary to list every object in the set. Instead, the rule that theobjects follow can be given in the braces. We can define a set by listing itselements or by describing its elements. The latter method is useful when workingwith large sets.

Exercises

Directions: Read each question below. Select your answer by clicking on its button. Feedbackto your answer is provided in the RESULTS BOX. If you make a mistake, rethink your answer,then choose a different button.

1.

Which of the following is the set of all suits in

a standard deck of playing cards?

R= [ace, two, three, four, five, six, seven, eight,nine, ten, jack, queen, king]

S= {hearts, diamonds, clubs, spades}

T= {jokers}

None of the above.

RESULTS BOX:

2.

Which of the following is the set of odd wholenumbers less than 10?

C= {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}


5/36

D= {0, 2, 4, 6, 8}

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

None of the above.

RESULTS BOX:

3.

Which of the following is the set of all oceanson earth?

G= {Atlantic, Pacific, Arctic, Indian, Antarctic}

E= {Amazon, Nile, Mississippi, Rio Grande,Niagara}

F= {Asia, Africa, North America, South America,Antarctica, Europe, Australia}

All of the above.

RESULTS BOX:

4.

Which of the following is the set of all types of

matter?

X= {iron, aluminum, nickel, copper, gold, silver}

Y= {hydrogen, oxygen, nitrogen, carbon dioxide}

Z= {liquids, solids, gases, plasmas}

None of the above.

RESULTS BOX:

5. Jennifer listed the set of all letters inthe word library as shown below.What is wrong with this set?

A = {l, i, b, r, a, r, y}


6/36

A capital letter is used to represent thisset.

It uses curly braces.

It uses commas.

The objects in this set are not unique.

RESULTS BOX:

Basic SetNotation

Unit15>

Lesson2 of 14

Problem 1: Mrs. Glosser asked Kyesha, Angie andEduardo to join the new math club. Afterschool they signed up and becamemembers. They wrote about it on thechalkboard using set notation:

P= {Kyesha, Angie and Eduardo}

When Angie's mother came to pick her up,she looked at the chalkboard andasked: What does that mean?

Solution: Let Pbe the set of all members in the mathclub.

An object that belongs to a set is called an element (or a member) of that set. We use specialnotation to indicate whether or not an element belongs to a set, as shown below.

Symbol Meaningis an element of

is not an element of

For example, given the set A = {1, 2, 3, 4, 5}, we can write 1 A, which stands for 1 is an

element of set A. We can also write 7 A, which stands for 7 is not an element of set A. Let'slook at some more examples of this.
http://www.mathgoodies.com/lessons/toc_unit15.htmlhttp://www.mathgoodies.com/lessons/toc_unit15.htmlhttp://www.mathgoodies.com/lessons/toc_unit15.htmlhttp://www.mathgoodies.com/lessons/toc_unit15.htmlhttp://www.mathgoodies.com/lessons/toc_unit15.html


7/36

Set Notation Meaning

A = {2, 4, 6, 8} 2 A 2 is an element of A

5 A5 is not an element of A

B= {a, e, i, o, u} e B e is an element of B

w B w is not an element of B

C= {1, 3, 5, 7, 9} 7 C 7 is an element of C

2 C 2 is not an element of C

D= {-3, -2, -1, 0, 1, 2, 3} -2 is an element of D

One-half is not an element of D

Determine if the given item is an element of the set.

Set Item Is an element?

R= {2, 4, 6, 8} 10

S= {2, 4, 6, 8, 10} 10

D= {English alphabet} m

D= {English alphabet}

X= {prime numbers less than 10} 9

A = {even numbers} 8

Summary: An object that belongs to a set is called an element (or a member) of that set. Weuse special notation to indicate whether or not an element belongs to a set: (

),


8/36

Exercises


1.

Which of the following is true for set R?

R= {liquid, gas, solid, plasma}

gas R

solid R

liquid R

None of the above.

RESULTS BOX:

2.

Which of the following is true for set G?

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

5 G

7 G

3 G

All of the above.

RESULTS BOX:

3.

Which of the following is true for set B?

B= {US flag colors}

red B


9/36

blue B

white B

All of the above.

RESULTS BOX:

4.

Which of the following elements is not amember of set X?

X= {tiger, lion, puma, cheetah, leopard,cougar, ocelot}

cougar

bobcat

puma

tiger

RESULTS BOX:

5. Which of the following elements is not a memberof set A?

A = {states in the US}

Guam

Haiti

Philippines

All of the above.

RESULTS BOX:

Types ofSets

Unit 15>Lesson 3


10/36

of 14

We learned how to write sets using roster notation,as shown in examples 1 and 2.

Example1:

Let Rbe the set of all vowels in theEnglish alphabet. Describe this setusing roster notation.

Solution: R= {a, e, i, o, u}

Example2:

Let Sbe the set of all letters in theEnglish alphabet. Describe this setusing roster notation.

Solution: S= {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}

In example 2, there are 26 elements in set S. It would be easier to use a shortcut to list this set:

Example2:

Let Sbe the set of all letters in the English alphabet. Describe this set using rosternotation.

Solution: S= {a, b, c, ..., x, y, z}

The three dots are called an ellipsis. We use an ellipsis in the middle of a set as a shortcut forlisting many elements. Note that the number of elements in set Rand set Sis countable, Soeach of these sets is a finite set. A finite set has afinitenumber of elements. Let's examineanother type of set:

Example 3: Let Tbe the set of all whole numbers.

Solution: T= {0, 1, 2, 3, 4, 5, 6, ...}

In example 3, we used an ellipsis at the end of the list to indicate that the set goes on forever.Set Tis an infinite set. An infinite set is a set with aninfinitenumber of elements. It is notpossible to explicitly list out all the elements of an infinite set. Let's look at some more examplesof finite and infinite sets.

F I N I T E S E T S I N F I N I T E S E T S
http://popupwindow%28%27finite%27%29/http://popupwindow%28%27finite%27%29/http://popupwindow%28%27infinite%27%29/http://popupwindow%28%27infinite%27%29/http://popupwindow%28%27infinite%27%29/http://popupwindow%28%27infinite%27%29/http://popupwindow%28%27finite%27%29/


11/36

Description Roster Notation Description Roster Notation

A = {whole numbers between0 and 100}

A = {1, 2, 3, ..., 97,98, 99}

W= {even wholenumbers}

W= {0, 2, 4, 6, 8, ...}

B= {primary colors}B= {red, blue,yellow}

X= {atoms in theuniverse}

X= {atom1, atom2,

atom3,...}

C= {prime numbers less than12}

C= {2, 3, 5, 7, 11} Y= {prime numbers} Y= {2, 3, 5, 7, 11, ...}

The ellipsis makes it easier to list both finite and infinite sets with roster notation. There are somesets that do not contain any elements at all, as shown below.

Example 4: Let Dbe the set of all weeks with 8 days.

Solution: D= {}

We call a set with no elements the null or empty set. It is represented by the symbol { } or .So D= {} or D= . Let's look at some more examples of empty sets.

Empty (Null) Sets

Description Notation

The set of dogs with sixteen legs. X= {}

The set of computers that are both on and off. Y= {}

The set of triangles with 4 sides. Z= {}

The set of months with 32 days. D=

The set of bicycles with no wheels. E=

The set of whole numbers that are odd and even. F=

Summary: An ellipsis is a shortcut used when listing sets with roster notation. A finite set hasa countable number of elements: An infinite has an infinite number of elements,such as the set of whole numbers, which goes on forever. We call a set with noelements the null or empty set.


12/36

Exercises


1.

What type of set is G?

G= {liquids, solids, gases, plasmas}

empty

finite

infinite

None of the above.

RESULTS BOX:

2.

What type of set is H?

H= {..., -3, -2, -1, 0, +1, +2, +3, ...}

empty

finite

infinite

None of the above.

RESULTS BOX:

3.

Which of the following sets are finite?

{vowels}

{days of the week}

{primary colors}

All of the above.


13/36

RESULTS BOX:

4. Which of the following is an infinite set?

{integers}

{states in the US}

{alphabet}

None of the above.

RESULTS BOX:

5. Which of the following is an empty (null) set?

{tiger, lion, puma, cheetah, leopard, cougar, ocelot}

{cars with more than 20 doors}

{prime numbers between 1 and 100}

All of the above.

RESULTS BOX:

SetEquality

Unit 15>Lesson 4 of

14

Problem 1: Mrs. Glosser asked her class to write the set ofprimary colors using roster notation. Shereceived two different answers from two differentstudents as shown below. Which student usedthe correct notation?

Student Notation

Eduardo X= {red, yellow, blue}


14/36

Angie Y= {blue, red, yellow}

Solution: Both students used the correct notation.

The sets from problem 1 are equal, and we write X= Y. The equals sign (=) is used to show equality.Let's look at some more examples of setequality.

Example 1: Are sets A and Bequal?

Solution:A = {1, 3, 5, 7}

B= {3, 7, 1, 5}

Examine these sets closely to confirm that they areequal.Answer: A = B

Since A and Bcontain exactly the same number of elements, and the elements in both are thesame, we say that A is equal to B, and we write A = B. The order in which the elements appear inthe set is not important.

Example 2: Are sets Xand Yequal?

Solution:X= {a, e, i, o, u}

Y= {u, o, i, e, a}

Examine these sets closely to confirm that they areequal.

Answer: X= Y

Since Xand Ycontain exactly the same number of elements, we write X= Y. Remember that theorder in which the elements appear in the set is not important.

Example 3: Are sets Pand Qequal?
http://popupwindow%28%27equality%27%29/http://popupwindow%28%27equality%27%29/http://popupwindow%28%27equality%27%29/http://popupwindow%28%27equality%27%29/


15/36

Solution:P= {apples, oranges, bananas, pears}

Q= {oranges, pears, apples}

Examine these sets closely to confirm that they are notequal.

Answer: PQ

Since Pand Qdo not contain exactly the same elements, we say that Pis not equal to Q, and wewrite P Q.

Example 4: Let Rbe the set of all whole numbers less than 5, and let S= {4, 2, 0, 3, 1}. Are

sets Rand Sequal?

Solution:R= {whole numbers < 5}

S= {4, 2, 0, 3, 1}

Examine these sets closely to confirm that they areequal.

Answer: R= S

Example 5: Which of the following setsare equal?

C= {1, 2, 3}

D= {a, e, i, o, u}

E= {2, 4, 6, 8, 10}

F= {John, Jane, Joe}

G= {2, 3, 1}

H= {o, e, a, y, u}

J= {2, 4, 6, 8}

K= {Jane, Joe, John}

Answer: C= Gand F = K


16/36

In example 5, these sets are NOT equal: D Hand E J. Can you name other sets that are notequal?

Summary: Two sets are equal if they have the exact same number of elements, and theirelements are the same. The order in which the elements appear in the set is notimportant.

Exercises

Directions: Read each question below. Select your answer by clicking on its button. Feedbackto your answer is provided in the RESULTS BOX. If you make a mistake, rethink your answer,

then choose a different button.

1.

Which of the following sets is equal to set P?

P= {Monday, Tuesday, Wednesday, Thursday,Friday}

W= {Thursday, Friday, Saturday, Sunday, Monday}

X= {Tuesday, Wednesday, Thursday, Friday,Saturday}

Y= {Thursday, Friday, Monday, Tuesday,Wednesday}

All of the above.

RESULTS BOX:

2.

Which of the following sets is not equal to

set H?

H= {5, 2, 1, 4, 3, 6}

M= {3, 2, 1, 4, 5, 6}

Q= {4, 1, 6, 2, 7, 3}

D= {1, 2, 6, 4, 5, 3}


17/36

None of the above.

RESULTS BOX:

3.

Let M= {0, 2, 4, 6, 8, 10}, and let N= {evennumbers < 10}. Which of the followingstatements is true?

Mis an infinite set.

M= N

MN

All of the above.

RESULTS BOX:

4.

Let X= {primary colors}, and let Y= {yellow,blue, red}. Which of the following statementsis true?

X= YY=

Xis an infinite set.

None of the above.

RESULTS BOX:

5. Let A = {}, and let B= . Which of the followingstatements is true?

A is an infinite set and Bis a finite set.

A is a finite set and Bis an infinite set.

A is null and B has one element.


18/36

A = B.

RESULTS BOX:

VennDiagrams

Unit 15>Lesson 5

of 14

Until now, we have examined sets using set notation. We know from previous lessons that thefollowing conventions are used with sets:

Capital letters are used to denote sets. Lowercase letters are used to denote elements of sets. Curly braces { } denote a list of elements in a set.

Another way to look at sets is with a visual tool called a Venn diagram, first developed by JohnVenn in the 1880s. In aVenn diagram, sets are represented by shapes; usually circles or ovals.The elements of a set are labeled within the circle. Let's look at some examples.

Example 1: Given set Ris the set of counting numbers less than 7. Draw and label a Venndiagram to represent set Rand indicate all elements in the set.

Analysis: Draw a circle or oval. Label it R. Put the elements in R.

Solution

Notation: R= {counting numbers < 7}

Example2:

Given set Gis the set of primary colors. Draw and label a Venn diagram to representset Gand indicate all elements in the set.

Analysis: Draw a circle or oval. Label it G. Put the elements in G.


19/36

Solution:

Notation: G= {primary colors}

Example

3:

Given set Bis the set of all vowels in the English alphabet. Draw and label a Venn

diagram to represent set Band indicate all elements in the set.

Analysis: Draw a circle or oval. Label it B. Put the elements in B.

Solution:

Notation: B= {vowels}

In each example above, we used a Venn diagram to represent a given set pictorially. Venndiagrams are especially useful for showing relationships between sets, as we will see in theexamples below. First, we will use a Venn diagram to find the intersection of two sets. Theintersection of two sets is all the elements they have in common.

Example 4: Let X= {1, 2, 3} and Let Y= {3, 4, 5}. Draw and label a Venn diagram to showthe intersection of sets X and Y.

Analysis: We need to find the elements that are common in both sets. Draw a picture oftwo overlapping circles. Elements that are common to both sets will be placed inthe middle part, where the circles overlap.


20/36

Solution:

Explanation: The circle on the left represents set Xand the circle on the right representsset Y. The shaded section in the middle is what they have in common. That istheir intersection.

The Venn Diagram in example 4 makes it easy to see that the number 3 is common to bothsets. So the intersection of Xand Yis 3. This is what Xand Yhave in common. The intersection

of Xand Yis written as and is read as"X intersect Y". So Intersection means "X andY". In example 5 below, we will find the union of two sets. The union of two sets is the setobtained by combining the elements of each.

Example 5: Let X= {1, 2, 3} and Let Y= {3, 4, 5}. Draw and label a Venn diagram to represent theunion of these two sets.

Analysis: To find the union of two sets, we look at all the elements in the two sets together.

Solution:

Explanation: Any element in X, Y, or in their intersection is in their union. So Xunion Yis {1, 2, 3,4, 5}. Both circles have been shaded to show the union of these sets.

The union of two sets is written as and is read as "X union Y". It means "X or Y". Let'scompare intersection and union.

Intersection Union

written as


21/36

read as X intersect Y X union Y

meaning of X and Y X or Y

Look for the elements in common to both combine all elements

The examples in this lesson included simple Venn diagrams. We will explore this topic in more depthin the next few lessons. We will also learn more about intersection and union in this unit.

Summary: We can use Venn diagrams to represent sets pictorially. Venn diagrams areespecially useful for showing relationships between sets, such as the intersectionand union of overlapping sets.

ExercisesDirections: Read each question below. Select your answer by clicking on its button. Feedbackto your answer is provided in the RESULTS BOX. If you make a mistake, rethink your answer,then choose a different button.

1.

Which of the following is represented by the Venndiagram below?

{A}

A = {odd numbers between 0 and 10}

A = {even numbers between 0 and 10}

None of the above.

RESULTS BOX:


22/36

2.

Which of the following is represented by the Venn

diagram below?

{B}

B= {hearts, diamonds, clubs, spades}

B= {jacks, queens, kings, aces}

None of the above.

RESULTS BOX:

3.

Which of the following is represented by the Venndiagram below?


23/36

P= {2, 4, 6, 8, 10}

Q= {6, 9}

PQ

All of the above.

RESULTS BOX:

4.

Which of the following is the correct roster notation

for set X?

X= {2, 3, 5, 6, 7}

X= {2, 3, 5, 7, 11}

X= {2, 3, 5, 7, 11, 15}

None of the above.

RESULTS BOX:


24/36

5. Which of the following relationships is shown by the Venndiagram below?

XY

XY

X= Y

All of the above.

RESULTS BOX:

Subsets Unit 15>Lesson 6 of14

Example 1: Given A = {1, 3, 4} and B= {1, 2, 3, 4, 5},what is the relationship between these sets?

We say that A is a subset of B, since everyelement of A is also in B. This is denoted by:

AVenn diagramfor the relationship betweenthese sets is shown to the right.

Answer: A is a subset of B.
http://www.mathgoodies.com/lessons/toc_unit15.htmlhttp://www.mathgoodies.com/lessons/toc_unit15.htmlhttp://popupwindow%28%27venn_diagram%27%29/http://popupwindow%28%27venn_diagram%27%29/http://popupwindow%28%27venn_diagram%27%29/http://popupwindow%28%27venn_diagram%27%29/http://www.mathgoodies.com/lessons/toc_unit15.html


25/36

Another way to define a subset is: A is a subset of Bif every element of A is contained in B. Bothdefinitions are demonstrated in the Venn diagram above.

Example 2: Given X= {a, r, e} and Y= {r, e, a, d}, what isthe relationship between these sets?

We say that Xis a subset of Y, since everyelement of Xis also in Y.This is denoted by:

A Venn diagram for the relationship betweenthese sets is shown to the right.

Answer: Xis a subset of Y.

Example 3: Given P= {1, 3, 4}and Q= {2, 3, 4, 5,6}, what is therelationshipbetween thesesets?

We say that Pisnot a subsetof Qsince notevery elementof Pis notcontained in Q. Forexample, we can

see that 1 Q.The statement "P isnot a subset ofQ"is denoted by:

Note that thesesets do have someelements incommon. The


26/36

intersection ofthese sets is shownin the Venndiagram to theright.

Answer: Pis not a subsetof Q.

The notation for subsets is shown below.

Symbol Meaning

is a subset of

is not a subset of

Example 4: Given A = {1, 2, 3, 4, 5} and B= {3, 1, 2, 5, 4},what is the relationship between A and B?

Analysis: Recall that the order in which the elementsappear in a set is not important. Looking at theelements of these sets, it is clear that:

Answer: A and Bare equivalent.

Definition: For any two sets, if A B and B A, then A = B. Thus A and Bare equivalent.

Example 5: List all subsets of the set C= {1, 2, 3}.

Answer:Subset Comment

D= {1} List all possible combinations of elements...


27/36


28/36

{x, y, z}. How many arethere?

Subsets

D= {x}

E= {y}

F= {z}

G= {x, y}

H= {x, z}

J= {y, z}

K= {x, y, z}

Answer: There are eight subsets ofthe set R= {x, y, z}.

{1, 2, 3, 4}. How many arethere?

Subsets

D= {1} M= {2, 4}

E= {2} N= {3, 4}

F= {3} O= {1, 2, 3}

G= {4} P= {1, 2, 4}

H= {1, 2} Q= {1, 3, 4}

J= {1, 3} R= {2, 3, 4}

K= {1, 4} S= {1, 2, 3, 4}

L = {2, 3}

Answer: There are 16 subsets ofthe set C= {1, 2, 3, 4}.

In example 6, set Rhas three (3) elements and eight (8) subsets. In example 7, set C has four(4) elements and 16 subsets. To find the number of subsets of a set with n elements, raise 2 tothe nth power: That is:

The number of subsets in set A is 2n

, where n is the number of elements in set A.

L e s s o n S u m m a r y

Subset

A is a subset of Bif every element of A is contained in B. This is denoted

by A B.

Equivalent Sets For any two sets, if A B and B A, then A = B.

Null set The null set is a subset of every set.

Sets and subsets Any set contains itself as a subset. This is denoted by A A.

Proper Subsets

If A B, and AB, then A is said to be a proper subset of Band it is

denoted by A B.

Number of SubsetsThe number of subsets in set A is 2n , where n is the number of elements inset A.


29/36

Exercises

Directions: Read each question below. Select your answer by clicking on its button. Feedback

to your answer is provided in the RESULTS BOX. If you make a mistake, rethink your answer,then choose a different button.

1.

Which of the following is a subset of set G?

G= {d, a, r, e}

X= {e, a, r}

Y ={e, r, a}

Z= {r, e, d}

All of the above.

RESULTS BOX:

2.

Which of the following statements is true?

{vowels} {consonants}

{consonants} {vowels}

{vowels} {alphabet}

None of the above.

RESULTS BOX:

3.

Which of the following is NOT a subset of set A?

A = {2, 3, 5, 7, 11}

B= {3, 5, 2, 7}

C= {2, 3, 7, 9}

D= {7, 2, 3, 11}

All of the above.

RESULTS BOX:


30/36

4.

How many subsets will the set below have?

T= {Monday, Tuesday, Wednesday, Thursday,Friday}

5

10

32

None of the above.

RESULTS BOX:

5. IfR= {whole numbers < 5} and S= {4, 2, 0, 3, 1}, thenwhich of the following statements is true?

R= S

Rhas more elements than S.

Sis null.

None of the above.

RESULTS BOX:

UniversalSet

Unit 15>Lesson 7

of 14

In previous lessons, we learned that a set is a group of objects, and that Venn diagrams can be

used to illustrate both set relationships and logical relationships.

Example 1: Given A = {1, 2, 5, 6} and B= {3, 9}, what is the relationship between these sets?

A and Bhave no elements in common. This relationship is shown in the Venndiagram below.


31/36

Answer: A and Bhave no elements in common. These sets do not overlap.

In example 1, A and Bhave no elements in common. (Each set is shaded with a different colorto illustrate this.) Therefore, it is logical to assume that there is no relationship between these

sets. However, if we consider these sets as part of a larger set, then there is a relationshipbetween them. For example, consider the single-digit numbers 1 through 9: If {1, 2, 3, 4, 5, 6, 7,8, 9} is our larger set, then A and Bare part of that set. Thus A and Bare each a subset of thislarger set, called the Universal Set.

Definition:A Universal Set is the set of all elements under consideration, denoted by capital .All other sets are subsets of the universal set.

Example 2: Given = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 5, 6} and B= {3, 9}, draw a Venndiagram to represent these sets.

Answer:

Think of a Universal set is the "big picture" It includes everything under consideration, oreverything that is relevant to the problem you have. In example 2, and . Notethat subsets A and Bdo not overlap: These sets aredisjoint. The procedure for creating a Venn
http://popupwindow%28%27disjoint%27%29/http://popupwindow%28%27disjoint%27%29/http://popupwindow%28%27disjoint%27%29/http://popupwindow%28%27disjoint%27%29/


32/36

diagram is as follows;

1. Draw a rectangle and label it U to represent the universal set.2. Draw circles within the rectangle to represent the subsets of the universe. Label the

circles and write the relevant elements in each circle.3. Write the remaining elements outside the circles but within the rectangle.

Let's look at some more examples.

Example 3: Given = {whole numbers less than 10}, P= {multiples of 3 less than 10}and Q= {even numbers less than 10}, draw a Venn diagram to represent thesesets.

Answer:

In example 3, subsets Pand Qare overlapping.

Example 4: Given = {whole numbers}, R= {primes numbers less than 12} and S= {evenprimes}, draw a Venn diagram to represent these sets.


33/36

Answer:

In example 4, Sis contained within R. This is due to the fact that the number 2 is the only evenprime. In addition, the universal set isinfinite, since the set of whole numbers goes on forever.Accordingly, we did not include any remaining whole numbers outside the circles and within therectangle.

Example 5: Given = {animals}, X= {dogs} and Y= {cats}, draw a Venn diagram torepresent these sets.

Answer:

In example 5, subsets Xand Ydo not overlap. Below is a word problem that you may findinteresting.
http://popupwindow%28%27infinite%27%29/http://popupwindow%28%27infinite%27%29/http://popupwindow%28%27infinite%27%29/http://popupwindow%28%27infinite%27%29/


34/36

Example 6: In a class of 10 students, some students were selected for the school band,some were selected for the school chorus, some were selected for both, andthe rest were selected for neither. Given = {Sam, Kyesha, Derek, Lorrie,Robin, Ral, Shirley, Nathan, Chris, Dana}, Band= {Sam, Lorrie, Ral,Derek} and Chorus= {Robin, Derek, Kyesha}, draw a Venn diagram to

represent these sets.

Answer:

In example 6, Bandand Chorusare overlapping sets. In addition, Band and Chorusare each asubset of the universal set, which is all the students in the class.

In this lesson, we examined several examples of universal sets with Venn diagrams. In someexamples, the sets overlapped and in some they did not. Also included were examples in whichone set was contained within the other.

Summary: A universal set is a set containing all elements of a problem under consideration,denoted by capital . A universal set includes everything under consideration, oreverything that is relevant to the problem you have. If the universal set containssets A and B, then and .

Exercises

Directions: Read each question below. Select your answer by clicking on its button. Feedbackto your answer is provided in the RESULTS BOX. If you make a mistake, rethink your answer,


35/36

then choose a different button.

1.

If G= {-9,

-8,

-7,

-6} and H= {

+8,

+2,

+7,

+4}, then which

of the following is the universal set?

= {fractions}={integers}

= {irrationals}

All of the above.

RESULTS BOX:

2.

If X= {Asia, Africa, North America, South America,Antarctica, Europe, Australia}, and Y= {Atlantic,Pacific, Arctic, Indian, Antarctic}, then which of thefollowing could be the universal set?

={oceans}

={countries}

={world}

All of the above.

RESULTS BOX:

3.

If = {whole numbers}, M= {even numbers} and N={odd numbers}, then which of the followingstatements is true?

M N =

All of the above.

RESULTS BOX:


36/36

4.

If = {polygons} and A= {quadrilaterals), thenwhich of the following sets does not overlapwith A,and is also is a part of ?

B ={rectangles}

C ={triangles}

D ={parallelograms}

None of the above.

RESULTS BOX:

5.

If = {whole numbers less than 40} and P= {1, 4, 9,16, 25, 36}, then which of the following sets overlapswith P, and is also a part of ?

Q={factors of 36}

R={multiples of 4}

S= {even primes}

None of the above.

RESULTS BOX:

introductions to sets

Documents