Types of Set | Sets || Mathematics in the Modern World

Summary

This educational video script introduces various types of sets in set theory, including finite, infinite, universal, equivalent, empty, unit, disjoint, overlapping, and subsets. It explains cardinality, the concept of equal and equivalent sets, and how to determine if a set is a subset or proper subset of another. The script uses examples to illustrate these concepts, making it easier for viewers to understand the fundamentals of set theory.

Takeaways

• 📚 There are various types of sets including definite, infinite, universal, equivalent, empty, unit, disjoint, overlapping, and subsets.
• 🔢 A finite set is one where the number of elements is countable, like the set of counting numbers less than or equal to 12.
• 🌈 Examples of finite sets include the colors of the rainbow and counting numbers up to a certain limit.
• 🚫 A set like the set of triangles with four sides is an example of an empty set, which has no elements.
• ∞ Infinite sets have elements that cannot be counted, such as the set of all positive integers.
• 📈 Cardinality refers to the count of elements in a set, denoted as n(S) for set S.
• 🔄 Equal sets have the exact same elements, regardless of the order, while equivalent sets have the same number of elements but may have different elements.
• ❌ Disjoint sets have no elements in common, whereas overlapping sets share at least one element.
• 🌍 The universal set contains all elements under consideration and can include elements from multiple other sets.
• 👥 A subset is a set where all its elements are also elements of another set, known as the superset.
• 🔑 The formula to determine the number of subsets of a set with 'n' elements is 2^n.

Q & A

What are the different kinds of sets discussed in the script?

The script discusses definite set, infinite set, universal set, equivalent set, empty set, unit set, disjoint set, overlapping set, and subset.

What is a finite set?

A finite set is a set whose elements are either empty or countable, meaning the number of elements can be counted.

Can you provide an example of a finite set from the script?

Yes, an example of a finite set is the set of counting numbers less than or equal to 12, which includes the numbers 1 through 12.

What is an infinite set?

An infinite set is a set whose elements cannot be counted, meaning there is no way to determine the exact number of elements in the set.

What is the cardinality of a set?

The cardinality of a set refers to the number of elements that belong to that set, denoted as n(S) where S is the given set.

How is the cardinality of a set determined?

The cardinality of a set is determined by counting the number of elements within the set.

What is an empty set?

An empty set, sometimes called a void or null set, is a set with no members or elements, and its cardinality is zero.

What is a unit set?

A unit set, also known as a singleton, is a set that contains only one element.

How are equal sets defined?

Two sets A and B are said to be equal if they have the same elements, regardless of the arrangement of the elements.

What is the difference between an equal set and an equivalent set?

Equal sets have the same elements, while equivalent sets have the same number of elements (cardinality) but not necessarily the same elements.

What is a universal set?

A universal set is the set of all elements under discussion, including all other given sets within that context.

How are disjoint sets defined?

Two sets are disjoint if they have no elements in common.

What is the difference between a disjoint set and an overlapping set?

A disjoint set has no common elements, while an overlapping set has at least one element in common.

What is a subset?

A subset is a set where each element of one set (A) is also an element of another set (B).

What is a proper subset?

A proper subset is a subset where the subset is not equal to the original set, meaning it has fewer elements.

How can you determine the number of subsets of a given set?

The number of subsets of a set can be determined using the formula 2^n, where n is the number of elements in the set.

Outlines

00:00

📚 Introduction to Different Kinds of Sets

The script introduces various types of sets in set theory, including definite, infinite, universal, equivalent, empty, unit, disjoint, overlapping, and subsets. It emphasizes the importance of understanding the basic concepts of sets before delving into more complex topics. The paragraph provides examples of finite sets, such as the set of counting numbers less than or equal to 12, and explains the concept by listing its elements and counting them. It also humorously points out the impossibility of a triangle with four sides, highlighting the importance of accurate definitions in mathematics.

05:02

🔢 Understanding Finite and Infinite Sets

This section delves into the concepts of finite and infinite sets. A finite set is defined as one whose elements are countable, exemplified by sets of colors of the rainbow and counting numbers up to 12. Infinite sets, on the other hand, are those with elements that cannot be counted, such as the set of all positive integers, natural numbers, and multiples of three. The paragraph also introduces the term 'cardinality,' which refers to the number of elements in a set, and provides examples to illustrate how to determine the cardinality of different sets.

10:08

🔄 Equal Sets, Equivalent Sets, and Cardinality

The script explains the difference between equal sets and equivalent sets. Equal sets have the same elements, regardless of the arrangement, while equivalent sets have the same number of elements, or cardinality. Examples are given to illustrate these concepts, such as comparing sets of letters and numbers. The paragraph also discusses the concepts of the empty set (denoted by the symbol Ø) and the unit set, which contains only one element. It clarifies that the empty set is not equivalent to a unit set with a different element.

15:08

🌐 Universal Sets and Their Applications

This part of the script defines the universal set, denoted by a capital letter U, as the set containing all elements under consideration. It provides examples to illustrate how to identify the universal set in different contexts, such as a set of letters from the English alphabet or a set of rational numbers. The universal set serves as a reference point for discussing other sets and their relationships.

20:11

🔗 Overlapping and Disjoint Sets

The script differentiates between overlapping and disjoint sets. Overlapping sets share at least one element in common, while disjoint sets have no elements in common. Examples are used to demonstrate these concepts, such as comparing sets of vowels and numbers. The paragraph also introduces the concept of a subset, where one set is entirely contained within another set, and the notation used to represent this relationship.

25:13

📉 Subsets and Proper Subsets

This section explains the concept of subsets and proper subsets. A subset is a set where all its elements are also elements of another set, while a proper subset is a subset that is not equal to the original set. The script provides examples to show how to determine if one set is a subset or a proper subset of another. It also mentions that the empty set is always a subset of any given set and concludes with a formula for calculating the number of subsets a set can have, which is 2 raised to the power of the number of elements in the set.

📈 Conclusion and Q&A

The script concludes by summarizing the discussion on the different kinds of sets and their identification. It invites viewers to ask questions in the comments if they have any, promising to address them. The presenter thanks the audience for watching and signs off with a positive note.

Keywords

💡Finite Set

A finite set is a collection of elements where the number of elements is countable. This means that it is possible to list all the elements or count them without reaching an infinite number. In the video, finite sets are exemplified by a set of counting numbers less than or equal to 12, which includes the numbers 1 through 12, and a set of colors of the rainbow, which includes red, orange, yellow, green, blue, indigo, and violet.

💡Infinite Set

An infinite set is a collection of elements that cannot be counted because there is no finite number of elements. This concept is central to the video's discussion of sets with uncountable elements. Examples given in the script include the set of positive integers, natural numbers, and multiples of three, all of which are infinite because they continue indefinitely.

💡Cardinality

Cardinality refers to the number of elements in a set. It is a fundamental concept in set theory that helps to quantify the size of a set. In the video, cardinality is used to distinguish between different types of sets, such as finite and infinite sets. For instance, the cardinality of a set of even numbers greater than five but less than 15 is five, as it includes the numbers 6, 8, 10, 12, and 14.

💡Equal Set

Equal sets are two sets that have the exact same elements, regardless of the order of those elements. This concept is important for understanding set equality. In the video, an example of equal sets is given where set A and set C have the same elements, thus making them equal sets.

💡Equivalent Set

Equivalent sets are two sets that have the same number of elements, or cardinality, even if the elements themselves are different. This concept is crucial for comparing the sizes of sets. The video provides an example where set B and set D have different elements but the same cardinality, making them equivalent sets.

💡Empty Set

The empty set, sometimes called the null set, is a set with no elements. It is denoted by the symbol '∅'. The video explains that the cardinality of the empty set is zero, as there are no elements to count. An example from the script is set A, which is defined as a set of triangles with four sides, an impossible condition, thus making it an empty set.

💡Unit Set

A unit set, also known as a singleton, contains exactly one element. This concept is important for understanding the simplest non-empty set. In the video, set B is described as a set of even prime numbers, which only includes the number 2, making it a unit set with a cardinality of one.

💡Universal Set

The universal set is a set that contains all elements under consideration within a given context. It serves as a reference point for defining other sets. The video explains that the universal set is denoted by a capital letter U and includes all elements being discussed. An example provided is the set of letters from the English alphabet, which serves as the universal set for discussing subsets like vowels or consonants.

💡Disjoint Set

A disjoint set is a set that has no elements in common with another set. This concept is important for understanding the relationships between sets. In the video, set B (vowels) and set C (numbers from one to five) are described as disjoint sets because they have no common elements.

💡Overlapping Set

An overlapping set is one that shares at least one element with another set. This is the opposite of a disjoint set and is important for understanding the intersection of sets. The video gives the example of set A and set B, which both contain the letter 'a', making them overlapping sets.

💡Subset

A subset is a set whose elements are all members of another set, known as the superset. If every element of set A is also an element of set B, then A is a subset of B. The video explains that if set A is a subset of set B but not equal to B, then A is a proper subset of B. An example from the script is set C, which is a proper subset of set A because all elements of C are in A, but A contains additional elements.

Highlights

Introduction to different kinds of sets

Definition of a finite set and examples

Explanation of infinite sets with examples

Concept of cardinality in sets

How to determine the cardinality of a set

Definition and example of equal sets

Definition and example of equivalent sets

Explanation of empty set and unit set

Definition and example of a universal set

Difference between disjoint and overlapping sets

Definition and example of a subset

Explanation of proper subset

Formula to determine the number of subsets

Examples of subsets for a given set

Conclusion of the discussion on types of sets

Invitation for questions from the audience

Transcripts

good day everyone

i hope we are doing fine today we will

learn about the different kinds of set

but if you haven't watched the

lecture about the introduction of set

you can click the link from the

description below

so what are the different kinds of set

we have definite set

infinite set

set

the universal set

equivalent set

empty set

unit set disjoint set overlapping set

and the subset

so what is a finite set

if we need set is a set whose element is

empty or countable

which means that we can count

the number of elements from the

given set

so

nothing belong

elements

given

we have here the examples of infinite

set so we have set s set a and set b

okay let's have the first set which is

the set s

so set s is that the set of x

such that x is a counting number less

than or equal to 12. so we need to list

down the elements

that are belong on the set s so let's

have set s is equal to the set of

counting numbers less than or equal to

12 so we have 1

2

3

4

5

6

7

8

9

10

11.

should we include the 12

yes because from the

definition less than or equal to

so we will include the

12.

now

can we count the number of elements from

the set s

yes we can easily count the number of

elements so we have one two three four

five six seven eight nine ten eleven and

twelve okay

let's have the second example set a

set is that

the set of x such that x are the colors

of the rainbow

so we all know that we can easily count

the colors of the rainbow

and mirandin

so we have the

red

orange

yellow

green

blue

indigo

violet

okay so

that uh these are the elements of this

set e we have the red orange shade of

green blue indigo and violets roy g beef

next the last one is we have

b

set b which is set b is the set of x

such as

such that x is a triangle with four

sides

okay set b dow is a triangle

with four sides

so do we have do we have a triangle that

has four sides of course none

malata young triangle

for size because triangles are three

sided

plane figure so we have

the

[Music]

the second one is we have the infinite

set

it is a set whose elements cannot be

counted so if the

infinite set is we can count the number

of elements

while infinite set is that we cannot

count the number of elements of any

given set we have here the examples of

infinite set so set a is a set of x such

that x is a positive integer so we all

know that we have we can we can't count

the number of elements that

define from this set a because there's a

lot of positive integers so we have set

a

is equal to the set of

one

two

three

four

five

six

and so on

and so forth

yeah nothing

okay next set b is the set of x such

that x is a natural numbers so we have a

lot of natural numbers also so set b

the set of 1

2

3

4

5 6

so on and so forth

yeah

and the last one we have the set c is

the set of x such that x is a multiples

of three

so indeed

multiples of three which are

we have three

six 9

12 15 18

[Music]

and so on and so forth

so these three

three sets class are some examples of

infinite set wherein we can count the

number of elements okay

before we proceed with the next type of

set class let us define first what does

cardinality means

cardinality of any given set class means

we talk about the number of elements

that are belong to that given set

so for cardinality we denote it as n of

s

where in the s the class represent any

given set if the given set is set a then

we write it as the cardinality of set a

is we have

n of a

so any letter that is used to represent

a

set so we have here three

examples of set and let's determine the

cardinality of this set so cardinality

which means all we have to do is to

count the number of elements of the

given set so for the set a we have 0 2 4

6 8

10 so if you will to count that is n of

a

is equal to one two three four five six

so

six

next the second one set b is the set of

x such that x is even numbers greater

than five but less than 15.

so if we will to list down the elements

from the set b or if you can do it

mentally okay long so if you will do

this down we have greater than 5 which

is even which are even numbers

so we have 6

8

10

12

14

okay

less than 15 last ha so

n of b now is we have

one two three four five so we have

five

and the last one set c is the set of x

such that x is an odd number less than

ten so add number less than 10

so we have

1

3

5

7

and

9.

so if we were to count one two three

four five so the cardinality of c is

five so that's how to

get the cardinality of any given set

next we have the equal set and the

equivalent set equal sets two sets a and

b are said to be equal if and only if

they have the same elements

so we write it as a equals b

which means the elements yuma elements

are set a

i makikit

b regardless of the arrangement of the

elements while equivalent set class two

sets are equivalent if and only if they

have the same number of elements which

means we are talking about the

cardinality

of two sets here so if the cardinality

of set a is the same with the

cardinality of set b

then they are equivalent set regardless

set b

now let's have the example of equal set

and the equivalent set

let's have the set a and set c

new class

the letters or the elements from set a

and c are the same

unanna

word as martin while young setsi is the

jumbo blancha therefore

set a and set c are

equal set because they have the same

elements regardless in the jumbo

elements no given set

now let's have the equivalent command as

you can see from the set b and set d

they are different elements mankind

elements but the cardinality of set b

and set d are equal

which are five

therefore set b and set d are equivalent

set

next one we have the empty set and the

unit set so empty set we sometimes call

it as void or now set which is a set

with no members which means no elements

and the cardinality of the empty set is

zero of course because

and we denote it denoted at by the

symbol

fee okay we call this symbol as fee or

sometimes we use the bracket in bracket

long

is a set with only one element

which means

we call it as unit set or

singleton

of the empty set and the unit set we

have two sets here a and b for the set a

we have the set of x such that x is a

triangle with four sides so i've

mentioned already this set it's a

previous uh

example not in kanina where in we all

know that we don't have a triangle that

has four sides kasikapak first sides ang

tawak dong

we can write it as set a

is empty by using the symbol or

bettering set a

is empty

okay pero remember class

encounter

nanganito

so the cardinality of this set a is one

element

it's the empty set

so this one class this one is not

equivalent with this

and also with this

sword please uh

note this one that this is not

equivalent with the empty with the

symbol that we are using for the empty

set

okay let's have the set b

set b is the set of x such that x is an

even prime number

okay so we list down the elements of the

set b we have

so on a young even prime number annoying

prime numbers that are even

okay so

we have 2

2 is an even prime number

uh even the prime number

therefore set b is a

unit or single tone set which means the

cardinality

is

one

is

element

unit set or single tone

set okay

next let's have the universal set so

universal set so we denote the universal

set as capital letter u

wherein it is the set of all elements

under discussion so it means that

yunkabu and nanset that is the universal

set so we have now an example of

universal

set so for the universal set is

it is the uh how we discuss or describe

yunkabu and nama given set so as you can

see we have six given set here wherein

we have a frequency

[Music]

so we have the set a b and c so the

universal set of these three sets are

the set

of

letters

from the

english

alphabet

so the universal set is the set of

letters from the english alphabet

hindi include class

letters

general description

are the set of letters from the english

alphabet is a b c d and e are part of

the english alphabet yes

is the vow are the vowels are part of

the english alphabet yes

are the letters that um

are the letters from my name christine

are part of the english alphabet

yes therefore the universal set of these

three set here are the set of letters

from the english alphabet they are part

okay how about the second the

these three sets are here we have the

set d e and f so what can you observe

as you can see we have

uh negative

integers here tapas my positive that was

my zero toposmo fraction

so if you are thinking the set of

rational numbers

then you are correct bucket hindi real

numbers

because real numbers are rational and

irrational

positive integers marine negative

integers therefore the universal set of

set b and f are

the set of

rational

numbers

okay

so that that is how to write a universal

set

universal universe

so that is universal

next let's have the difference between

this joint set and overlapping set

this joint set is that two sets are

disjoined

if and only if they have no elements in

common ebooks

while overlapping set

two sets are overlapping if and only if

they have at least one element in common

if it's a begin

the two sets are over overlapping kappa

meron silang

is a lang at least one which means one

or more let's have an example here we

have three sets again the a b and c and

let's see

what sets here are the this join set and

what sets here are the overlapping set

so

do you think a

and

b

are

overlapping set

okay if your answer is yes

then you are correct why because as you

can see class from the elements of a

meron tayo

m-a-r-t-i-n

and from the elements of b miranda a e i

therefore a and b are

overlapping

set

how about b and c

okay if your answer is that b and c are

disjoint set therefore you are correct

why because as you can see from the

elements of b we have the vowels and the

element of c are

numbers from one to five

and from the definition of the disjoint

set

two sets are disjoint if and only if

they have no command when unpacked

therefore b and c are disjoint set how

about a and c

again if your answer is disjoint you are

correct because a and c

are have no common elements so letters

okay

so that that is the difference between

the

overlapping and disjoint set let's have

now the subset so for two sets a and b

we say that a

is a subset of b so this symbol stands

for subset so a is a subset

of b if each element of a sola

or

yes

elements down a

is

element

b

if a is a subset of b class

but

a is not equal to b

again in this

lms

b

then we write

a is a proper subset of b

and b

elements

subset b or a is a proper subset of

b okay

so that is a difference between subset

and proper subset this example here so

we have set a to set g so tignan

subset

proper subset

so do you think class

dot

b is a subset of a

do you think c

is a subset of a

or do you think g

is a subset of a

or do you think e

is a subset of a

or

f

is a subset of a subset

a is a subset of b

if the elements of a

is part of the elements of b so d o

b is a subset of a

are the elements of b are part of the

elements of a

if your answer is yes then this is true

that b is a subset of a casi

b a

c a makita

okay next

c is a subset of a

what are the elements of c

a and b

a and b elementary

next g is a subset of a or

uh

the given set can be a subset of itself

class

next

therefore this is

wrong

then

f is a subset of a

annoying f nothing empty

okay

so remember class that

empty set

is always a subset of any given set

again

empty set is always a subset of any

given set therefore f is a subset of a

is correct

okay that is how subset works from all

the given set anonymous

subset this one class

c

is a proper subset of a y

because they are not equal

a b

a i a b c therefore c is a proper subset

of

a weathering b b is a proper subset of

elements

then this one is not a sub proper subset

of the set a

always remember that we can say that two

sets are proper subset

okay so that is the difference of subset

and proper substitution no class that we

have a formula

in determining the number of subset of

any given set

let's say we have the set a

the set a

whose elements are a

b

and c

using the formula two raised to n class

we can now determine the number of

subsets of this set

so how many how many elements do we have

from the set a we have three

so 2 raised to 3

that is 8 therefore

there are 8 subsets from this set

okay

which are

the empty set

this set of a

this set of b

set of c

set of

a

and b

set of

b

and c

set of

a

and c

and the set of itself which is a b

and c

again

so that this set here are the subset of

set a tama

this fee is a subset of set a this a is

a subset of set set a

this set is a subset of set a

paper

so

indeed

so that is how to determine the number

of elements or i mean the number of

subsets of a given set using the formula

2 raised to n

that ends our discussion about the

different kinds of set i hope 19

on how to identify what type of set are

given if you have some question just hit

the

comment below and

i will entertain your question thank you

for watching god bless everyone bye bye