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Mathematics Division, IMSP, UPLB

Set Relations

Learning Objectives:

Upon completion you should be able to identify set relations such as

• equality

• subset and superset

• equivalence


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 _______.


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


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


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


Set Relations Subsets and Supersets


B G

S

U

Set Relations Subsets and Supersets


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?

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.


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


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


Set Relations Subset


SUBSETS OF SET J

PROPER IMPROPER

Empty set

Set J

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


2. A A (Reflexive Property)

4. If A B then B A

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


Set Relations

A=B if and only if A B and B A.

ALTERNATIVE DEFINITION OF EQUALITY OF SETS


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}


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?




1-1 Correspondence


1

20

3

a

b

c

THEY ARE EQUIVALENT


Does the following exhibits 1-1 Correspondence? Are they equivalent?


10

20

38

a

b

c

d

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

Example


between

the set of days in a week and

the set of months in a year?

NO

THEY ARE NOT EQUIVALENT

Example


between

the set of even counting numbers and

the set of odd counting numbers?

YES

THEY ARE EQUIVALENT

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


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}


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


QUESTION:

IN SET THEORY,

•Is countable and finite the same?

•Is uncountable and infinite the same?

CARDINALITY

Cardinality of a set is a

measure of the size or the

“number of elements” of

the set.

What is the cardinality of { }?

COUNTING, 1-1 CORRESPONDENCE

AND CARDINALITY

1

2

3

a

b

c

SET OF NATURAL/

COUNTING

NUMBERS SET J

COUNTING, 1-1 CORRESPONDENCE

AND CARDINALITY

The cardinality of set J is

|J| = n(J) = 3

CARDINALITY AND

SET EQUIVALENCE

Two sets are equivalent if

they have the same

cardinality.

A set where you can have

1-1 correspondence with a

subset of natural/counting

numbers is countable.

Otherwise, it is

uncountable.

COUNTABLE SET

Question: Is the set of

positive even integers

countable?

COUNTABLE SET

1 2 3 4 …

COUNTABLE SET

2 4 6 8…

YES! This is called countably infinite!

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!


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,…}


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!


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.


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)

Summary

Infinite sets can be countable or uncountable.

All uncountable sets are infinite sets. But not all infinite sets are uncountable.


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

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.

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.

The concept of INFINITY is

mysterious. You may read

some articles about this

concept on the internet…

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