Types of Set | Sets || Mathematics in the Modern World
Summary
TLDRThis 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
- đ There are various types of sets including definite, infinite, universal, equivalent, empty, unit, disjoint, overlapping, and subsets.
- đą A finite set is one where the number of elements is countable, like the set of counting numbers less than or equal to 12.
- đ Examples of finite sets include the colors of the rainbow and counting numbers up to a certain limit.
- đ« A set like the set of triangles with four sides is an example of an empty set, which has no elements.
- â Infinite sets have elements that cannot be counted, such as the set of all positive integers.
- đ Cardinality refers to the count of elements in a set, denoted as n(S) for set S.
- đ 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.
- â Disjoint sets have no elements in common, whereas overlapping sets share at least one element.
- đ The universal set contains all elements under consideration and can include elements from multiple other sets.
- đ„ A subset is a set where all its elements are also elements of another set, known as the superset.
- đ 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
đ 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.
đą 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.
đ 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 Ă) 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.
đ 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.
đ 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.
đ 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.
đ 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.
Mindmap
Keywords
đĄFinite Set
đĄInfinite Set
đĄCardinality
đĄEqual Set
đĄEquivalent Set
đĄEmpty Set
đĄUnit Set
đĄUniversal Set
đĄDisjoint Set
đĄOverlapping Set
đĄSubset
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
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