ako, si , ay nangangakong magsisipag mag-aral hindi lang para sa...
TRANSCRIPT
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Ang aking kontrata:
Ako, si ______________, ay
nangangakong magsisipag mag-aral
hindi lang para sa aking sarili kundi
para rin sa aking pamilya, para sa
aking bayang Pilipinas at para sa
ikauunlad ng mundo.
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Mathematics Division, IMSP, UPLB
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Set Relations
Learning Objectives:
Upon completion you should be able to identify set relations such as
• equality
• subset and superset
• equivalence
Mathematics Division, IMSP, UPLB
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Set Relations
Two sets A and B are equal if and only if they have the same elements.
Example:
Let A = {1, 3, 5, 7, 9} and B = {3, 5, 1, 9, 7}.
Equal Sets
Are all the elements in A also in B?
Are all the elements in B also in A?
YES! Therefore A and B have the same elements.
Sets A and B are _______.
Mathematics Division, IMSP, UPLB
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Set Relations
If sets A and B are equal, we write A=B.
Otherwise, we write A B.
Example:
Let C = {1, 2, 3, 4} and
D = {1, 2, 2, 3, 4, 4}.
Are all the elements in C also in D?
Are all the elements in D also in C?
YES! Therefore C = D.
Equal Sets
Mathematics Division, IMSP, UPLB
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Set Relations
REMEMBER:
1. IN A SET, IT IS CUSTOMARY TO LIST AN ELEMENT ONLY ONCE.
2. IN A SET, THE ORDER OF LISTING THE ELEMENTS DOES NOT MATTER.
3. TWO SETS ARE EQUAL IF AND ONLY IF THEY HAVE THE SAME ELEMENTS.
Equal Sets
Mathematics Division, IMSP, UPLB
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Set Relations
If all elements of set A are also elements of set B, we say, A is a subset of B (or B is a superset of A).
Example: Let S be the set of all students in this room.
Let B be the set of all boys in this room.
Let G be the set of all girls in this room with age less than 16.
Is B a subset of S? Is G a subset of S?
Subsets and Supersets
Mathematics Division, IMSP, UPLB
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Set Relations Subsets and Supersets
Mathematics Division, IMSP, UPLB
B G
S
U
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Set Relations Subsets and Supersets
Mathematics Division, IMSP, UPLB
TIME TO THINK!
1. Is U always a superset?
2. Is a set a subset of itself?
3. Is a set a superset of itself?
4. Do you think { } is a subset of any set?
5. Do you think { } is a superset of any set?
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Set Relations
Subset
If A is a subset of B, we write A B.
B S and G S.
However, B is not a subset of G. Why?
In this case, we write B G.
In our previous example,
S is the set of all students in this room.
B is the set of of all boys in this room.
G is the set of all girls in this room with age less than 16.
Mathematics Division, IMSP, UPLB
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Set Relations
Suppose A is a non-empty set. If A B and A B, then we call A a proper subset of B.
If A = {, } and B = {,,,} then A is a proper subset of B and we may write A B or A B.
Subset
Mathematics Division, IMSP, UPLB
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Set Relations
There are only two improper subsets. The empty set and the set itself.
If A = {, } and B = {,,,} then
{} and A are improper subsets of A and we write {} A and A A.
{} and B are improper subsets of B and we write {} B and B B.
Subset
Mathematics Division, IMSP, UPLB
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Set Relations Subset
Mathematics Division, IMSP, UPLB
SUBSETS OF SET J
PROPER IMPROPER
Empty set
Set J
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Set Relations Always True, Sometimes True or False: Let A, B, and C be sets.
1. A A
3. If A B then B A
5. If A B and B C then A C (Transitive Property)
6. {} A
7. {} {}
8. A U
Subset
Mathematics Division, IMSP, UPLB
2. A A (Reflexive Property)
4. If A B then B A
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Set Relations
Determine if proper or improper subset of {1,2,3,4,5}:
1. {1,2}
2. {4}
3. {1,2,3,4,5}
4. {2,3,4}
5. {}
Subset
Mathematics Division, IMSP, UPLB
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Set Relations
A=B if and only if A B and B A.
ALTERNATIVE DEFINITION OF EQUALITY OF SETS
Mathematics Division, IMSP, UPLB
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Set Relations Set Equivalence
What can you observe about the following pairs of sets?
A = {1, 2, 3, 4, 5}
B = {a, e, i, o,u}
C = {guava, melon, avocado}
D = {do, re, mi}
Mathematics Division, IMSP, UPLB
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Two sets are in 1-1 correspondence if it is possible to pair each element of A with exactly one element of B, and each element of B with exactly one element of A. It follows that A and B have the same size or number of elements.
When two sets A and B are in 1-1 correspondence, we say they are equivalent and we write A B.
Thus, in our example, A B and C D.
Is A C? Why?
Set Relations Set Equivalence
Mathematics Division, IMSP, UPLB
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Set Relations Set Equivalence
1-1 Correspondence
Mathematics Division, IMSP, UPLB
1
20
3
a
b
c
THEY ARE EQUIVALENT
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Set Relations Set Equivalence
1-1 Correspondence
Mathematics Division, IMSP, UPLB
1
20
3
a
b
c
THEY ARE EQUIVALENT
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Set Relations Set Equivalence
Does the following exhibits 1-1 Correspondence? Are they equivalent?
Mathematics Division, IMSP, UPLB
10
20
38
a
b
c
d
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Set Relations Set Equivalence
Does the following exhibits 1-1 Correspondence? Are they equivalent?
Mathematics Division, IMSP, UPLB
10
20
38
a
b
c
d
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Example
Is there a one-to-one correspondence
between the set of days in a week and
the set of counting numbers from 2 to 8?
M T W Th F Sa Su
2 3 4 5 6 7 8
YES
THEY ARE EQUIVALENT
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Example
Is there a one-to-one correspondence
between
the set of days in a week and
the set of months in a year?
NO
THEY ARE NOT EQUIVALENT
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Example
Is there a one-to-one correspondence
between
the set of even counting numbers and
the set of odd counting numbers?
YES
THEY ARE EQUIVALENT
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Set Relations
Time to think:
1. Are all equal sets equivalent?
2. Are all equivalent sets equal?
3. Can a set be equivalent to any of its subsets?
4. Can a set be equal to any of its subsets?
Set Equivalence
Mathematics Division, IMSP, UPLB
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Set Relations
Exercise
For each of the sets listed below, tell which are
equivalent and which are also equal.
1. The set of distinct letters in the word
“katakataka”
2. The set {a,k,t,k}
3. The set of distinct letters in the word “tatak”
4. The set {k,t,a}
5. The set {k,a,r}
Mathematics Division, IMSP, UPLB
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Set Relations
Summary
In this section, we learned
• When two sets are equal or not
• When a set is a subset or superset of another
• When two sets are equivalent or not
Mathematics Division, IMSP, UPLB
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QUESTION:
IN SET THEORY,
•Is countable and finite the same?
•Is uncountable and infinite the same?
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CARDINALITY
Cardinality of a set is a
measure of the size or the
“number of elements” of
the set.
What is the cardinality of { }?
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COUNTING, 1-1 CORRESPONDENCE
AND CARDINALITY
1
2
3
a
b
c
SET OF NATURAL/
COUNTING
NUMBERS SET J
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COUNTING, 1-1 CORRESPONDENCE
AND CARDINALITY
The cardinality of set J is
|J| = n(J) = 3
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CARDINALITY AND
SET EQUIVALENCE
Two sets are equivalent if
they have the same
cardinality.
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A set where you can have
1-1 correspondence with a
subset of natural/counting
numbers is countable.
Otherwise, it is
uncountable.
COUNTABLE SET
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Question: Is the set of
positive even integers
countable?
COUNTABLE SET
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1 2 3 4 …
COUNTABLE SET
2 4 6 8…
YES! This is called countably infinite!
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FINITE AND INFINITE SET
A set is finite if it has a cardinality equal to a counting/natural number.
Example: n(J)=3
All finite sets are countable!
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FINITE AND INFINITE SET
A set is finite if your counting ends.
A set is finite if after listing all the elements, there is a last element.
Example:{a,b,c,d,e}
Counterexample:{1,2,3,4,5,…}
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FINITE AND INFINITE SET
A set is infinite if it is not finite.
Example: The set of natural/counting numbers {1,2,3,…} has infinitely many elements, hence it is an infinite set. But, it is countable!
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FINITE AND INFINITE SET
Example: The set of positive even integers does not have cardinality equal to a natural/counting number, so it is infinite. But it can have a 1-1 correspondence with the set of natural/counting number so it is countable.
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FINITE AND INFINITE SET
Example of uncountable infinite set:
The set of real numbers
(because we cannot have a 1-1 correspondence between the set of reals and the set of counting numbers)
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Summary
Infinite sets can be countable or uncountable.
All uncountable sets are infinite sets. But not all infinite sets are uncountable.
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FINITE AND INFINITE SET
Exercise: Determine if FINITE or INFINITE, and if COUNTABLE or UNCOUNTABLE
1)Set of points in a circle
2)Set of counting numbers between 1 and 1023
3)Set of real numbers between 0 and 1
4) The set of all sands in Boracay beach
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FYI
The cardinality of the set of natural/counting numbers is ℵ0 (aleph-null).
The cardinality of the set of real numbers is ℵ1 (aleph-one) or c (for continuum).
Note: ℵ0 and ℵ1 are not real numbers.
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TRIVIA
Can a set be equivalent to one of its proper subset?
YES! When would this happen?
If the set is infinite. Can you give an example?
N~E.
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The concept of INFINITY is
mysterious. You may read
some articles about this
concept on the internet…