1-6 Set Theory Warm Up Warm Up Lesson Quiz Lesson Quiz Lesson Presentation Lesson Presentation.

1-6 Set Theory

Warm UpWarm Up

Lesson QuizLesson Quiz

Lesson PresentationLesson Presentation

1-6 Set Theory

Warm Up

1.

2.

3. 25

4. –6

Write all classifications that apply to each real number.

5.

59

5

710

rational, repeating decimal

irrational

rational, terminating decimal, integer,whole, natural

rational, terminating decimal, integer

rational, terminating decimal

1-6 Set Theory

MA.912.D.7.1 Perform set operations such as union and intersection, complement, and cross product.Also MA.912.D.7.2, MA.912.A.10.1.

Sunshine State Standards

1-6 Set Theory

Perform operations involving sets.Use Venn diagrams to analyze sets.

Objectives

1-6 Set Theory

set elementunionintersectionempty setuniversecomplementsubsetcross product

Vocabulary

1-6 Set Theory

A set is a collection of items. An element is anitem in a set. You can use set notation to representa set by listing its elements between brackets. The set F of riddles Flore has solved is F = {1, 2, 5, 6}. The set L of riddles Leon has solved is L = {4, 5, 6}.

The union of two sets is a single set of all the elements of the original sets. The notation F L means the union of sets F and L.

Union

1 265 4

3Set F Set L

F L = {1, 2, 4, 5, 6}Together, Flore and Leonhave solved riddles 1, 2,4, 5, and 6.

1-6 Set Theory

The intersection of two sets is a single set that contains only the elements that are common to the original sets. The notation F ∩ L means the intersection of sets F and L.

Intersection

1 265 4

3set F set L

F L = {5, 6}Flore and Leon have both solved riddles 5 and 6.

∩

The empty set is the set containing no elements. It is symbolized by or {}.

1-6 Set Theory

Writing Math

In set notation, the elements of a set can bewritten in any order, but numerical sets areusually listed from least to greatest without repeating any elements.

1-6 Set Theory

Find the union and intersection of each pair of sets.

Additional Example 1A: Finding the Union and Intersection of Sets

A = {5, 10, 15}; B = {10, 11, 12, 13}

10155 11Set A Set B

12 13

A U B = {5, 10, 11, 12, 13, 15}

To find the union, list every element that lies in one set or the other.

1-6 Set Theory

Additional Example 1A Continued

A ∩ B = {10}

To find the intersection, list the elements common to both sides.


A = {5, 10, 15}; B = {10, 11, 12, 13}

10155 11Set A Set B

12 13

1-6 Set Theory


Additional Example 1B: Finding the Union and Intersection

A is the set of whole number factors of 15;

B is the set of whole number factors of 25.

A U B = {1, 3, 5, 15, 25}

A ∩ B = {1, 5}

Write each set in set notation.

To find the union, list all of the elements in either set.

To find the intersection, list the elements common to both sets.

A = {1, 3, 5, 15}B = {1, 5, 25}

1-6 Set Theory

Check It Out! Example 1a


A = {–2, –1, 0, 1, 2}; B = {–6, –4, –2, 0, 2, 4, 6}

A U B = {–6, –4, –2, –1, 0, 1, 2, 4, 6}

A ∩ B = {–2, 0, 2}



1-6 Set Theory

Check It Out! Example 1b

A is the set of whole numbers less than 10; B is the set of whole numbers less than 8.


A U B = {0, 1, 2, 3,4, 5, 6, 7, 8, 9}

A ∩ B = {0, 1, 2, 3, 4, 5, 6, 7}



Write each set in set notation.

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

B = {1, 2, 3, 4, 5, 6, 7}

1-6 Set Theory

The universe, or universal set, for a particular situation is the set that contains all of the elements relating to the situation. The complement of set A in universe U is the set of all elements in U that are not in A.

In the contest described on slide 6, the universe U is the set of all six riddles. The complement of set L in universe U is the set of all riddles that Leon has not solved.

Complement of L

1 265 4

3Set F Set L

Universe U

Complement of L = {1, 2, 3}. Leon has not solved riddles 1, 2, and 3.

1-6 Set Theory

Additional Example 2A: Finding the Complement of a Set

U is the set of natural numbers less than 10; A is the set of whole-number factors of 9.

Find the complement of set A in universe U.

A = {1, 3 ,9}; U = {1, 2, 3, 4, 5, 6, 7, 8, 9}

1

2

6543

Set A

Universe U

9

7

8Draw a Venn diagram to show the complement of set A in universe U

Complement of A = {2, 4, 5, 6, 7, 8}

1-6 Set Theory

U is the set of rational numbers; A is the set of terminating decimals.

Find the complement of set A in universe U.

Complement of A = the set of repeating decimals.

Additional Example 2B: Finding the Complement of a Set

1-6 Set Theory

Reading Math

Finite sets have finitely many elements, as in Example 2A. Infinite sets have infinitely many elements, as in Example 2B.

1-6 Set Theory

Find the complement of set A in universe U. U is the set of whole numbers less than 12; A is the set of prime numbers less than 12.

Check It Out! Example 2

{0, 1, 4, 6, 8, 9, 10}

1-6 Set Theory

One set may be entirely contained within another set. Set B is a subset of set A if every element of set B is an element of set A. The notation B A means that set B is a subset of set A.

1-6 Set Theory

Additional Example 3: Determining Relationships Between Sets

A is the set of positive multiples of 3, and B is the set of positive multiples of 9. Determine whether the statement A B is true or false. Use a Venn diagram to support your answer.

Set Bmultiplesof 9

Set A multiples of 3 that are not multiples of 9

Draw a Venn diagramto show these sets.

False; B A

1-6 Set Theory

Check It Out! Example 3

A is the set of whole-number factors of 8, and B is the set of whole-number factors of 12. Determine whether the statement A B = B is true or false. Use a Venn diagram to support your answer.

1Set A

248

Set B3

612

False; the element 8 ofset A, is not an elementof set B.

1-6 Set Theory

The cross product (or Cartesian product) of two sets A and B, represented by A B, is a set whose elements are ordered pairs of the form (a, b), where a is an element of A and b is an element of B. You can use a chart to find A B. Suppose A = {1, 2} and B = {40, 50, 60}.

A B = {(1, 40), (1, 50), (1, 60), (2, 40), (2, 50), (2, 60)}

2

1

Set B

Set A

60 50 40

(1,40)

(2,40)

(1,50)

(2,50)

(1,60)

(2,60)

1-6 Set Theory

The set C = {S, M, L} represents the sizes of cups (small, medium, and large) sold at a frozen yogurt shop. The set F = {V, B, P} represents the available flavors (vanilla, banana, peach). Find the cross product C F to determine all of the possible combinations of sizes and flavors.

Additional Example 4: Application

S M L

V

B

P

(S,V) (M,V) (L,V)

(S,B) (M,B) (L,B)

(S,P) (M,P) (L,P)

Set C

Set F

Make a chart to find the cross product.

Each pair represents one combination of flavors and sizes.

{(S, V), (S, B), (S, P), (M, V), (M,B), (M, P), (L,V), (L, B), (L, P)}; 9 possible combinations

C F =

1-6 Set Theory

Check It Out! Example 4The set MN = {M, N, MN} represents the blood groups in the MN system. Find ABO × MN to determine all of possible blood groups in the ABO × MN systems. N

M

(O, M)(O, N)(O, M) O

(AB,MN)(AB,N)(AB,M)AB

(B, MN)(B, N)(B, M) B

(A, MN)(A, N)(A, M) A

MN

Make a chart to find the cross product. Each pair represents one combination of ABO and MN blood groups.

ABO MN = {(A, M), (A, N), (A, MN), (B, M),(B, N), (B, MN), (AB, M), (AB, N), (AB, MN), (O, M), (O, N), (O, MN)}: 12 possible blood groups.

1-6 Set Theory

Standard Lesson Quiz

Lesson Quizzes

Lesson Quiz for Student Response Systems

1-6 Set Theory

Lesson Quiz: Part I

1. Find the union and intersection of sets A and B. A = {4, 5, 6}; B = {5, 6, 7, 8}

A U B = {4, 5, 6, 7, 8}; A ∩ B = {5, 6}

2. Find the complement of set C in universe U. U is the set of whole numbers less than 10;

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

1-6 Set Theory

D is the set of whole-number factors of 8, and E is the set of whole-number factors of 24. Determine whether the statement D E is true or false. Use a Venn diagram to support your answer.

Lesson Quiz: Part II

3.

true

Set D Set E

348

2

6

12

24

1

1-6 Set Theory

F G = {(–1, –2), (–1, 0), (–1, –2), (0, –2), (0, 0), (0, 2), (1, –2), (1, 0), (1, 2)}

Find the cross product F G.

F = {–1, 0, 1}; G = {–2, 0, 2}

4.

Lesson Quiz: Part III

1-6 Set Theory


1. A set is defined as:

A. a collection of items

B. a collection of elements

C. a union of items

D. a union of elements

1-6 Set Theory

2. The symbol means:

A. intersection

B. union

C. empty set

D. set notation


1-6 Set Theory

3. The intersection:

A. contains common elements

B. is the empty set

C. contains the union

D. contains uncommon elements


1-6 Set Theory

4. Find the intersection of the two sets.

A. A B = {1, 3, 4, 5, 6, 7}

B. A B = {2}

C. A B = {2}

D. A B = {1, 3, 4, 5, 6, 7}1

Set A

248

Set B3

612


1-6 Set Theory

5. Find the compliment of set A in universe U.

A. {2, 4, 6, 8}

B. {1, 3, 6, 7, 8}

C. {1, 3, 5, 7, 9}

D. {1, 3, 5, 7}

U = All whole-numbers less than 9A = All even numbers


1-6 Set Theory Warm Up Warm Up Lesson Quiz Lesson Quiz Lesson Presentation Lesson Presentation.

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