Transcript SETS

Look at the
following
illustrations..
pharmacist
laptop
policeman
sister
father
teacher
iwatch
engineer
mother
doctor
Describe what you see.
cellphone
ipod
Waccamaw
Andrews
Carvers Bay
Georgetown
Verizon, AT&T,
T-Mobile,
Sprint
A
B
Mouth, esophagus,
stomach, liver, pancreas,
small intestine, large
intestine, appendix,
rectum, anus
C
Red, Orange,
Yellow, Green,
Violet, Indigo, Blue
D
North America,
South America,
Europe, Asia, Africa,
Antarctica, Australia
E
A SET is a well-defined collection
of distinct objects.
(Usually denoted by CAPITAL letters.)
- A SET is well-defined if its contents can be
clearly be determined
Examples:
 Examples:
The countries that are The set of the states of
USA
part Rio Olympics
Examples:
2016
The
set
of
the
Examples:
SENIOR students of
The set of all real
WHS
numbers
Is the group of places a set?
How about group of boys?
Is the group of three best movies
a set?
“WELL - DEFINED SET – means it is
possible to determine whether a
particular object belongs to a particular
set.
Example: Tell whether each set is well
defined or not
1. The collection of all the Math subjects
offered in WHS
Well defined
2. The collection of all basketball players
not
3.The collection of all schools in Georgetown
Well defined
4. The collection of the best teachers
not
5. The collection of your favorite shirt
not
- The objects in a set are called
ELEMENTS
of the set.
()
P  {a, b}
bP
cP
“b is an element of P”
“c is not an element of P”
or 

Bugs Bunny 
____ Pokemon
 odd integer
24 ____
 Presidents of the US
George Bush _____
Chicago Bulls ____ NBA Teams

A set that contains no element, is
denoted by,
  or 
EXAMPLES:
- Set of WHS students taking 3 Math
subjects in a school year
- Set of grade 9 students in Discrete
Math
Indicate the set of natural numbers
that satisfies the equation x+2 = 0
Only the number -2 satisfies this
equation. Because -2 is not a natural
number, the solution set for this
equation is
  or 
Roster Form
Elements are listed
and separated by
commas.
Examples:
 A = {1, 2, 3, 4, 5}
Rule Form/Set Builder
Notation
Elements are
described as {x|x …}
x such that…
Examples:
A = {x| x is a nutaral
number less than 6}
A = {x|x  N and x < 6}
Example: Describe the following sets in rule form.
1. A = { 3, 5, 7, 9 }
A = {x|x is an odd number between 2 and 10}
2. B = {Monday, Tuesday, Wednesday, Thursday,
Friday}
B = {x|x is a weekday}
3. C = {square, rhombus, parallelogram,
rectangle, trapezoid}
C = {x|x is a quadrilateral}
4. D = {red, orange, yellow, green, blue,
indigo, violet}
D = {x|x is a color of the rainbow}
Example: Describe the following sets in roster
form.
1. L = {x|x is an even number between 0 and
10}
L = { 2, 4, 6, 8 }
2. E = { x|x is a prime number less than 20}
E = { 2, 3, 5, 7, 11,13, 17, 19 }
3. F = { x|x is a vowel in the word
mathematics}
F = { a, e, i }
4. G = { x|x is an integer}
G= { …-3, -2, -1, 0, 1, 2, 3, … }
Example: Describe the following sets in roster
form.
1. L = {x|x

N and 2 ≤ x ≤ 8}
L = { 2, 3, 4, 5, 6, 7 }
The CARDINAL NUMBER of a set
- The cardinal number of Set A,
symbolized by n(A), is the number of
elements in set A.
Examples: State the cardinal number of the given set.
1. E = { x|x is a vowel in the alphabet}
n(E) = 5
2. F = {0, 1, 2, 3, 4, 5, 6}
n(F) = 7
3. G = { }
n(G) = 0
4. H = {2, 4, 6, 8, 10, … }
n(H) = Cannot be determined
Set of the
Continents of
the World
___________
7
Set of
Rational
Numbers
___________
Cannot be
determined
Set of Even
Integers between
15 and 37
___________
11
Kinds of Sets
 Countable
number
of elements
 Examples:
 A = { x | x is a
consonant}
 B = {1,2,3,4,5}
 C = { x | x is a
Math teacher in
WHS}
Finite Sets
 The
counting of its
elements has no end.
 Examples:
 A = { x | x is an even
number}
 B = {5,10,15,20, ...}
 C = { x | x is a negative
number}
Infinite Sets
 All
elements are the
same
 Example:
If J = {2, 4, 6, 8, 10}
K = { x | x is an
even number
between 0 and 11}
then
J=K
Equal Sets
 Same
number of
elements
 Example:
If R = {c, u, e, t}
and
S = {e, l, o, v}
then
n(R) = n(S)
Equivalent Sets
Any sets that are equal must also be equivalent.
Not all sets that are equivalent are equal.
There is at least one
common element.

Example:
If P  5,6,7,8,9,19
and Q  4, 2,1,6,9,19
then P and Q are joint
sets.



Joint Sets
There is no common
element.
Example:
The sets of positive and
negative numbers are
disjoint.
Disjoint Sets
Universal Set
(symbolized by U)
- Is the totality of the elements under
consideration.
- Is a set that contains all the elements for any
specific discussion
If P= {1, 2, 3, 4, …} ,
N = {0} and
Q = {-1, -2, -3, …}
then the universal set is
U= {…-3, -2, -1, 0, 1, 2, 3,…} or
U= {x|x is an integer}
Class Work - August 23, 2016
I. Determine if the given set is well defined.
1. The collection of all easiest courses in WHS
2. The set of the elements in the periodic table
3. The collection of all successful professionals
4. The collection of all rational numbers.
5. The collection of letters in the alphabet
II. Describe the following sets in roster and rule form. State the
cardinality.
6. The collection of natural numbers
7. The collection of prime numbers between 10 and 50
8. The collection of multiples of 8 less than 40
9. The collection of square numbers greater than 1
10. The collection of months with 31 days
III. Tell whether the statement is sometimes true, always true or never true.
11. Equal sets are also equivalent sets.
12. If two sets are disjoint, then they are equivalent.
13. The cardinality of a finite set cannot be determined.
14. If two sets are equal then they are also joint sets.
15. The set of all the students in WHS in 2010 has a finite cardinal number.
I. Tell whether the statement is sometimes true,
always true or never true.
1. If two sets are equivalent then they are equal.
2. The set {1, 2} is equal to the set { 2,1}.
3. The set of decimals and fractions are joint sets.
4. Rational numbers combined with irrational numbers
create the universal set of real numbers.
II. Describe the following sets in roster and rule form. State
the cardinal number of each set.
5. The set of cube numbers greater than 1
6. The set of nonnegative odd integer less than 20
7. The set of natural numbers
III. Tell whether the given set/s is/are finite, infinite, equal,
equivalent, joint or disjoint. (each item may have more
than one answer)
8. A = { 2, 4, 6, 8, 10} and B = {1, 3, 5, 7,9}
9. A = { 0, 1, 2, 3, 4,…} and B = {1, 2, 3, 4, …}
10. A = { 1, 2, 4} and B = {2, 1, 4}
SUBSETS
Set A is a subset of Set B, denoted as A  B, if and only if
all the elements of Set A are also elements of Set B
Example:
If A = { 1, 2, 3} and B = {1, 2, 3, 4, 5}
then
A B
• Every set is a subset of itself.
A A
• An empty set is always a subset of every set.
Ø
A
Subsets
Determine whether set A is a subset of set B.
a. A = {hydrogen, gold, silver}
B = {hydrogen, gold, iron, silver}
b. A = {5, 10, 15, 20, 25, 30}
B = {5, 10, 15, 25}
c. A = {chocolate, vanilla, rocky road}
B = {rocky road, chocolate, vanilla}
Proper Subsets
Proper Subsets
Consider:
R = {1, 2, 3, 4, 5, 6, 7, 8, 9}
A = {1, 2, 5, 6}
B = {3, 9}
C = {0, 1, 2, 10}
Is B
RIs

R?R?
A R?

Is
C
Determine whether the following is
TRUE or FALSE.
* The number of subsets of a given set is
2n where n is the number of elements in a
set.
Example: List all the subsets of M = {2, 3, 4}
1. A = { 2 }
5. E = { 2, 4 }
2. B = { 3 }
6. F = { 3, 4 }
3. C = { 4 }
7. G = { 2, 3, 4 }
4. D = { 2, 3 }
8. H = { }
DOOR
PRICE
BOARD
STAFFORD,
JASMINE
CHARLTON
BARKER
BROWN
TABLE
PROJECTO
R
KEATING
GREENE
STAFFORD,
JOSHUA
RATTO
PRICE,
HUNTER
KOONTS
CROOKS
VELTRI
ROURK
TAYLOR
SANGREGO
RY
ROY
LOTTCHEA
SMALL
PLEASE FOLLWING THE SEATING CHART
Venn Diagram (by John Venn) is used to
represent relationships between sets.
U
A
10
2
1
3
4
B
5
8
6
7
9
1. Union of Sets A and B
A  B = set of all elements found in A or B, or both A
and B.
Example: A U B = { 1, 2, 3, 4, 5, 6, 7}
2. Intersection of Sets A and B
AB
U
= set of all
common
elements in
A and B.
A
10
2
1
3
4
5
B
8
6
7
9
Example: A ∩ B = { 4, 5 }
3. Complement of Set A
A'
= set of all elements in the Universal set but not
found in A
A'  6,7,8,9,10
3. Difference of Sets A and B
A  B = set of all
A B
B A
U
elements found
in A but not in B.
A
= { 1, 2, 3 }
10
2
1
3
4
5
B
6
8
7
9
= { 6, 7 }
Example: Let A = {1, 2, 3, 4} and let B = {3, 4, 5, 6} and C = { 6, 7,8}
Find:
A  B = { 1, 2}
A C = {1, 2, 3, 4}
B  A = { 5, 6}
B  C = { 3, 4, 5}
U
X
Y
5
1
6
2
3
9
7
8
Z4
Use the Venn Diagram to list the members of the
specified sets.
1. X  Y
4.  X  Y    X  Y 
7. Z '
2. Z  Y
5.  X  Z   Y  Z 
8. U  
3. X '
6. Y  Z
VIDEO PRESENTATION
BASIC
VENN
DIAGRAM
SHADING
Shade the region representing
each set.
1.A
U B U C
2.A U B ∩ C
’
3.A -(B ∩ C)
’
4.(B U C)- A
Consider the ff. sets. Give the elements
of the sets being described below.
E = {3, 4, 5, 6} , I = {2, 6, p} , J = {y, n, p, y}
1. What is the universal set?
U = {2, 3, 4, 5, 6, p, n, o, y}
2. E’ = {2, o, n, p, y}
3. E∩I ∩J = { }
4. (I U J)’ ∩ E = {3, 4, 5}
5. (J - I) U E = {3, 4, 5, 6, n, o, y}
6. (E U I) – (J ∩ I) = {2, 3, 4, 5, 6}
CLASS WORK
I. Use the Venn Diagram to list the members of the specified sets.
1.
2.
3.
4.
5.
6.
7.
A B
B C
 A  B  C
 A  B   A  C 
A'
A B
B C
A
1
3
7
8
0
5
U
B
6
11
II. List all the subsets of A = {1, 5, 9}.
C9
4
2
III. Shade the region representing each set.
1. A  B  C 
A
2.  A  B   A  C 
B
C
B
C
U
U
4. B'C
3. A'
A
U
A
A
B
C
U
B
C
Assignment: ½ crosswise
In a survey of 75 consumers, 12 indicated that they were
going to buy a new car, 18 said they were going to buy a
new refrigerator, and 24 said they were going to buy a
new washer. Of these, 6 were going to buy both a car and
a refrigerator, 4 were going to buy a car and a washer,
and 10 were going to buy a washer and a refrigerator.
One person indicated that she was going to buy all three
items.
Construct a Venn diagram, label your diagram
clearly. Use your diagram to answer the following
questions:
(a) How many were going to buy only a car?
(b) How many were going to buy only a washer?
(c) How many were going to buy only a refrigerator
(d) How many were going to buy a car and a washer but
not a refrigerator?
(e) How many were going to buy none of these items?