1. Warm Up
Lesson Presentation
Lesson Quiz

2. Warm Up Write all classifications that apply to each real number. 1.2.
3. 25
4. –6
Write all classifications that apply to each real number.
rational, repeating decimal
irrational
rational, terminating decimal, integer,
whole, natural
rational, terminating decimal, integer
rational, terminating decimal 5.

3. Sunshine State Standards
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.

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

5. Vocabulary set element union intersection empty set universe
complement
subset
cross product

6. Union A set is a collection of items. An element is an
item in a set. You can use set notation to represent
a 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 2 6 5 4 3
Set F
Set L
F L = {1, 2, 4, 5, 6}
Together, Flore and Leon
have solved riddles 1, 2,
4, 5, and 6.

7. 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 2 6 5 4 3
set 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 {}.

8. Writing Math
In set notation, the elements of a set can be
written in any order, but numerical sets are
usually listed from least to greatest without
repeating any elements.

9. Additional Example 1A: Finding the Union and Intersection of Sets
Find the union and intersection of each pair of sets.
A = {5, 10, 15}; B = {10, 11, 12, 13} 10 15 5 11
Set A
Set B 12 13
To find the union, list every element that lies in one set or the other.
A U B = {5, 10, 11, 12, 13, 15}

10. Additional Example 1A Continued
Find the union and intersection of each pair of sets.
A = {5, 10, 15}; B = {10, 11, 12, 13} 10 15 5 11
Set A
Set B 12 13
To find the intersection, list the elements common to both sides.
A ∩ B = {10}

11. Additional Example 1B: Finding the Union and Intersection
Find the union and intersection of each pair of sets.
A is the set of whole number factors of 15;
B is the set of whole number factors of 25.
A = {1, 3, 5, 15}
B = {1, 5, 25}
Write each set in set notation.
A U B = {1, 3, 5, 15, 25}
To find the union, list all of the elements in either set.
A ∩ B = {1, 5}
To find the intersection, list the elements common to both sets.

12. Check It Out! Example 1a
Find the union and intersection of each pair of sets.
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}
To find the union, list all of the elements in either set.
A ∩ B = {–2, 0, 2}
To find the intersection, list the elements common to both sets.

13. Check It Out! Example 1b
Find the union and intersection of each pair of sets.
A is the set of whole numbers less than 10;
B is the set of whole numbers less than 8.
A = {1, 2, 3, 4, 5, 6, 7, 8, 9}
B = {1, 2, 3, 4, 5, 6, 7}
Write each set in set notation.
To find the union, list all of the elements in either set.
A U B = {0, 1, 2, 3,4, 5, 6, 7, 8, 9}
To find the intersection, list the elements common to both sets.
A ∩ B = {0, 1, 2, 3, 4, 5, 6, 7}

14. 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.
Complement of L 1 2 6 5 4 3
Set F
Set L
Universe U
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, 2, 3}. Leon has not solved riddles 1, 2, and 3.

15. Additional Example 2A: Finding the Complement of a Set
Find the complement of set A in universe U.
U is the set of natural numbers less than 10;
A is the set of whole-number factors of 9.
A = {1, 3 ,9}; U = {1, 2, 3, 4, 5, 6, 7, 8, 9} 1 2 6 5 4 3
Set A
Universe U 9 7 8
Draw a Venn diagram to show the complement of set A in universe U
Complement of A = {2,
4, 5, 6, 7, 8}

16. Additional Example 2B: Finding the Complement of a Set
Find the complement of set A in universe U.
U is the set of rational numbers;
A is the set of terminating decimals.
Complement of A = the set of repeating decimals.

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

18. Check It Out! Example 2
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.
{0, 1, 4, 6, 8, 9, 10}

19. One set may be entirely contained within another set
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.

20. 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 B
multiples
of 9
Set A multiples of 3 that are not multiples of 9
Draw a Venn diagram
to show these sets.
False; B  A

21. 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. 1
Set A 2 4 8
Set B 3 6 12
False; the element 8 of
set A, is not an element
of set B.

22. 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}. 2 1
Set B
Set A 60 50 40
(1,40)
(2,40)
(1,50)
(2,50)
(1,60)
(2,60)
A  B = {(1, 40), (1, 50), (1, 60), (2, 40), (2, 50), (2, 60)}

23. Additional Example 4: Application
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. 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
{(S, V), (S, B), (S, P), (M, V), (M,B), (M, P), (L,V), (L, B), (L, P)}; possible combinations C
 F =
Make a chart to find the cross product.
Each pair represents one combination of flavors and sizes.

24. N M O B A Check It Out! Example 4
The 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
(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.

25. Lesson Quizzes
Standard Lesson Quiz
Lesson Quiz for Student Response Systems

26. 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}

27. Lesson Quiz: Part II 3.
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.
true
Set D
Set E 3 4 8 2 6 12 24 1

28. Lesson Quiz: Part III
Find the cross product F  G.
F = {–1, 0, 1}; G = {–2, 0, 2} 4.
F  G = {(–1, –2), (–1, 0), (–1, –2), (0, –2), (0, 0), (0, 2), (1, –2), (1, 0), (1, 2)}

29. Lesson Quiz for Student Response Systems
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

30. Lesson Quiz for Student Response Systems
2. The symbol  means:
A. intersection
B. union
C. empty set
D. set notation

31. Lesson Quiz for Student Response Systems
3. The intersection:
A. contains common elements
B. is the empty set
C. contains the union
D. contains uncommon elements

32. Lesson Quiz for Student Response Systems
4. Find the intersection of the two sets.
A. A  B = {1, 3, 4, 5, 6, 7} 1
Set A 2 4 8
Set B 3 6 12
B. A  B = {2}
C. A  B = {2}
D. A  B = {1, 3, 4, 5, 6, 7}

33. Lesson Quiz for Student Response Systems
5. Find the compliment of set A in universe U.
U = All whole-numbers less than 9
A = All even numbers
A. {2, 4, 6, 8}
B. {1, 3, 6, 7, 8}
C. {1, 3, 5, 7, 9}
D. {1, 3, 5, 7}