1-6 Set Theory Warm Up Warm Up Lesson Quiz Lesson Quiz Lesson Lesson Presentation 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.
Find the union and intersection of each pair of sets.
A = {5, 10, 15}; B = {10, 11, 12, 13}
10155 11Set A Set B
12 13
1-6 Set Theory
Find the union and intersection of each pair of sets.
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
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}
A ∩ B = {–2, 0, 2}
To find the union, list all of the elements in either set.
To find the intersection, list the elements common to both sets.
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.
Find the union and intersection of each pair of sets.
A U B = {0, 1, 2, 3,4, 5, 6, 7, 8, 9}
A ∩ B = {0, 1, 2, 3, 4, 5, 6, 7}
To find the union, list all of the elements in either set.
To find the intersection, list the elements common to both sets.
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
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
1-6 Set Theory
2. The symbol means:
A. intersection
B. union
C. empty set
D. set notation
Lesson Quiz for Student Response Systems
1-6 Set Theory
3. The intersection:
A. contains common elements
B. is the empty set
C. contains the union
D. contains uncommon elements
Lesson Quiz for Student Response Systems
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
Lesson Quiz for Student Response Systems
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
Lesson Quiz for Student Response Systems