Introduction to Sets || Mathematics in the Modern World
Summary
TLDRThis educational video explores the concept of sets in mathematics, using relatable examples like mobile game skins and kitchen utensils to illustrate how objects with similar properties are grouped. It explains the notation for sets and elements, introduces the roster and set builder methods for defining sets, and discusses interval notation. The video also covers cardinality, distinguishing between finite and infinite sets, and the null set. It concludes with a discussion on set equality, emphasizing that the order of elements and duplicates do not affect set equivalence.
Takeaways
- 😀 A 'set' is a collection of well-defined objects, often with similar properties, referred to as elements.
- 🎲 Sets are commonly denoted using uppercase letters, with elements enclosed in braces.
- 👥 The language of sets is used to study collections in an organized manner, such as grouping all kitchen utensils as a 'kitchen set'.
- 🔢 In mathematics, sets can be formed using numbers and their properties, like even numbers or integers.
- 📝 There are different ways to describe a set, including the roster method, which lists all members, and set builder notation, which describes the properties elements must have.
- 🌐 Interval notation is used to describe sets of real numbers within a certain range, using parentheses to indicate inclusion or exclusion of endpoints.
- 🔑 The cardinality of a set refers to the number of elements it contains and can be represented with specific symbols.
- 📉 Finite sets are countable, meaning all their elements can be listed, while infinite sets have elements that cannot all be listed.
- ❌ The null set is a unique set with no elements, often represented by a slash through a circle or empty braces.
- 🔀 Sets can be equal if they contain the same elements, regardless of the order in which they are listed.
Q & A
What does the term 'set' generally refer to in the context of the video?
-In the video, 'set' refers to a collection of well-defined objects or elements that often share similar properties or characteristics.
How are sets commonly denoted in mathematical notation?
-Sets are commonly denoted using uppercase letters, while the elements are enclosed within braces.
What is an example of a set given in the video?
-An example of a set given in the video is the set of 'Zojac skins' in the mobile legends game, which are based on the 12 fire signs of constellation.
What symbols are used to denote that an element is part of a set?
-The symbols used to denote that an element is part of a set are '∈' (element of) and '∉' (not an element of).
What is the roster method for describing a set?
-The roster method is a way to describe a set by listing all its members between braces.
How is a set described when its elements are too many to list?
-When the elements of a set are too many to list, the roster method can be used with ellipses to indicate the pattern, or set builder notation can be employed.
What is set builder notation and how is it used?
-Set builder notation is used to describe a set by stating the properties that the elements must have to be members of the set, rather than listing all the elements.
What is interval notation and how does it describe sets?
-Interval notation is used to describe sets of real numbers within a specified range, using parentheses to indicate whether the endpoints are included or not.
What is the cardinality of a set and how is it represented?
-The cardinality of a set is the number of elements it contains. It can be represented using the symbols '∣' or '∣∣' followed by the set name.
What is the difference between finite and infinite sets?
-Finite sets are sets with a countable number of elements that can be listed, while infinite sets have an unlimited number of elements that cannot be listed.
What is a null set and how is it represented?
-A null set is a set with no elements. It is represented with a circle and a slash or empty braces.
How are equal sets defined and what is the significance of the order of elements in determining set equality?
-Equal sets are sets that contain the same elements, regardless of the order of those elements. The order does not affect the equality of sets, as sets are unordered collections.
Outlines
🌟 Introduction to Sets
This paragraph introduces the concept of sets, which are collections of objects or elements with similar properties. It explains that sets can be used to group various things together, such as kitchen utensils, gaming characters' armor, or students in a school. The paragraph emphasizes that sets should be well-defined to identify their elements and are commonly denoted using uppercase letters with elements listed within braces. It provides examples of sets, including a set of skins in a mobile game and a set of numbers, and explains how to denote elements as part of or not part of a set. The language of sets is presented as a way to study collections systematically.
📚 Describing Sets Using Different Notations
This paragraph delves into various methods of describing sets, such as the roster method, where all members are listed within braces, and set builder notation, which characterizes elements by their properties. It also touches on interval notation for describing sets of real numbers within certain ranges. The paragraph provides examples for each method, including sets of integers, even numbers, and rational numbers. It clarifies the use of ellipses and parentheses in roster and interval notations to indicate inclusivity or exclusivity of certain values.
🔢 Cardinality and Types of Sets
The third paragraph discusses the cardinality of sets, which refers to the number of elements they contain. It explains how to determine the cardinality for finite sets and provides examples of counting elements in sets of letters and numbers. The paragraph also introduces the concepts of finite and infinite sets, giving examples of each, and discusses the null set, which contains no elements. It concludes with a brief explanation of equal sets, where two sets are considered equal if they have the same elements and cardinality, regardless of the order in which those elements are listed.
🎓 Conclusion and Call to Action
The final paragraph summarizes the video's content on sets and invites viewers to subscribe, like, and enable notifications for more educational video tutorials. It serves as a conclusion to the video, reminding viewers of the value of the information covered and encouraging them to stay engaged with the channel for future content.
Mindmap
Keywords
💡Set
💡Elements
💡Well-defined
💡Uppercase Letters
💡Braces
💡Roster Method
💡Set Builder Notation
💡Interval Notation
💡Cardinality
💡Finite and Infinite Sets
💡Null Set
💡Equal Sets
Highlights
Definition of a 'set' as a collection of objects with similar properties.
Sets can be denoted using uppercase letters, with elements enclosed in braces.
Examples of sets in everyday life, such as kitchen utensils and gaming console characters' armor sets.
Introduction to the concept of elements within a set.
Explanation of how to denote that an element is part of or not part of a set using symbols.
Demonstration of set notation with a specific example from the mobile game 'Mobile Legends'.
Discussion on the flexibility of sets to include seemingly unrelated elements.
Description of the roster method for listing set members.
Use of ellipses in roster method when elements are too numerous to list.
Introduction to set builder notation as an alternative to roster method.
Explanation of interval notation for describing sets of numbers.
Examples of how to write sets using roster and interval notations.
Definition and calculation of cardinality in sets.
Distinction between finite and infinite sets.
Identification of null sets and their representation.
Concept of equal sets and the irrelevance of element order in set equality.
Encouragement for viewers to subscribe and engage with the content for more tutorials.
Transcripts
hello my name is ram and welcome to
another video of
matoklasan set may mean anything
sitcom said set score in a valuable game
last set of a band and many more
but set is a term that we often hear and
use to group
objects together often but not always
the objects in a set have similar
properties
for instance all the things that we use
to prepare
and cook food are part of the kitchen
set
when you complete gears for characters
in a gaming console
we call them armor set
even people can be grouped together to
form a set
like all the students who are currently
enrolled in a particular school
make up a set if we group
all the people with the name seth they
are called
set of sets interesting right
the language of sets is a means to study
such collections in an organized fashion
so a set is a collection of well-defined
objects
these objects are called elements
remember that a set should always be
well-defined to identify the elements
it is common for sets to be denoted
using uppercase letters
while the elements are enclosed using
braces
for example let a be the set of zojack
skins in the mobile legends game
for those who don't know skins are like
costumes and armor sets for a particular
character in the famous mobile legend
game
the skins are mostly based on galaxies
and stars so the zojac skins are based
on the 12 fire signs of constellation
so set a contains sagittarius
capricorn aquarius pisces aristorus and
so on
we write these symbols to denote that a
is
an element of the set a and
these symbols to denote that a is not
an element of the set a
in the previous example libra is an
element of a
because libra is in the set while king
though another skin in the game is not
an element of
set a because there is no king here
inside the set in mathematics we use
numbers and their properties to form a
set
in this example c is a set containing 1
2
3 and 4. 1 and 2 are element of
set a 6 is not an element of a
because there is no 6 within the set
z is a variable that could be any value
so it's not an element of set a
unless you see it here although sets are
usually used to group together elements
with common properties
there is nothing that prevents a set
from having seemingly unrelated elements
for instance in this example is a set
containing
4 elements a 1 smiley icon
and mask there are several ways to
describe a set
one way is to list all the members of a
set when this
is possible we use a notation where all
members of the set are listed between
braces
this way of describing set is known as
the roster method
for example the set v of all vowels in
the english alphabet can be written as
set v
containing a e i o u
the set e of even counting numbers less
than
10 can be expressed by set e containing
2
4 6 8. notice that 10 is not included
here
because it's not less than this number
sometimes the roster method is used to
describe a set without listing all its
members
when the elements of a set are too many
list
some elements first then use ellipses
but you need to make sure that the
general pattern of the elements
is obvious
[Music]
ellipses
ellipses can be placed in different
order depending on the pattern of the
set
for instance in this example the set of
integers
is written using side-by-side ellipses
in the set of whole numbers you could
start with 0
and end it with an ellipsis
in the set of negative integers since
this set is
extending to negative infinity then it
should start with
an ellipsis
another way to describe a set is to use
set builder notation
we characterize all those elements in
the set by stating the property or
properties they must have to be
members the set x
such that x is an even number less than
or equal to 10
is written in set builder notation
because instead of listing all the
elements
we use the properties of all the
possible elements like
8 6 4 and 2.
now how about these examples set o
is x such that x is an odd positive
integer
less than ten can you give me at least
one element for this one
yes it could be what three
what else seven is possible
and so on and this set q
is a set containing x such that x
is any rational number like
three over five
and we can also use interval notation if
we want to describe sets
do you recall interval notation when a
and b
are real numbers with a less than b we
can write this
following interval notations so if you
can see here
the bracket a to b it means that
x is an element of any real number
such that x is greater than or equal to
a
but less than or equal to b
in this interval we can see here that 2
is beside the open parenthesis
2.1 is possible because 2.1 is a real
number
greater than two federal
five federal seven point
eighty nine puerto ri
now
how about in this interval yes
4 is included in the set
5 is also included 6.33
is also included but 8
is not because this is an open
parenthesis
guys remember that if you happen to see
this kind of interval it's called closed
interval
and if you happen to see both
parentheses
it's an open interval example for the
close interval
we can have one third
to let's say nine
for the open interval it could be eleven
to
one hundred
now how about we try these examples
write the set of integers greater than
negative 4 but
less than 59 using the roster method
using the proster method we will start
with
negative 3 because it's the next highest
integer to negative 4.
we cannot use decimal numbers here
because we
only need integers so the next one
is negative 2. the next is
negative one and then followed by zero
now since we established our pattern
we can now use ellipses
three dots and up to
50
why 58 of course because the numbers
should be less than 59
in the next example we need to write the
set of positive real numbers between
one half and 100 over seven in interval
notation
i think this is easy because when we say
between
one half and one hundred over seven are
not
included in the set so i'm going to use
open
and close parentheses on both sides
and this is now our final answer
but of course if you want to define this
using set builder notation
it's like this x
is an element of r such that x is 1
greater than one half
but less than 100
over 7 that's 100 okay
so notice that i did not write any equal
sign here because these are
open and close parentheses
now how about this let us write this set
in set builder notation form ready for
the answer
okay the first thing that we need to do
is to specify that
x is an element of the set of integers
why because 3.1 3.2
are not allowed in the set so we need to
specify that they
are all integers such that x is a
multiple
of three because these numbers
are all multiple of three
but of course they should be less than
or
equal to 15.
the cardinality of set refers to the
number of elements
in a set so we can use either of these
two symbols to represent the cardinality
of
set a now why don't we try these
examples
let the set a be a set containing x such
that
x is a lowercase english alphabet letter
okay so all we need to do here is to
count the alphabets
in english right so there are about what
26 so the cardinality of a is
26. how about this set
all you need to do is to count the
elements one two
three four five six so the answer here
is
six how about this one
x is an element of this means
set of positive integers so we will only
consider the positive integers
and x is an even number
less than 10. oh so this should be on
okay so how many all right we start with
two
and then what how about four
six eight but 10 is not
included right because it should be less
than 10.
are we missing something no we cannot
include zero because we only need
positive integers so
the answer here is four
how about this last set
x is an element of positive integers
again
such that x is an even number less than
or
equal to 10. etiquette
because this time 10 is included in the
set
so the answer here is 5
good job and speaking of cardinality
we can only count the elements of finite
sets because finite sets
are sets which either has no elements or
has elements which could all be possibly
listed
down they are countable countable yeah
let r be the set of natural numbers less
than ten can we count them all
of course we could start with one two
three four five six
up to nineteen okay so that's an example
of a finite set
and infinite sets are sets whose
elements cannot be
listed unlimited set of all real numbers
so this is an example of an infinite set
and we also have the null set
these are sets with no elements
us a circle and a slash or braces like
this
example set of positive integers between
1 and 10 that are divisible by 13 is
there a number between 1 and 10 that is
divisible by 13 and none
so this is an example of a null set
number two set of integers between two
and three
nope there are no numbers
or integers between 2 and 3
because when we say integers 2.1 is not
an integer 2.2 is not an integer
2 and 3 are consecutive positive
integers and guys do you know that we
also
have equal sets let a and b be sets if
both a and b
have the same elements then a is equal
to b
in number one set a here is equal to
set b on the other side why
guys instead the order is not important
so though this is one two three four and
this is one four two three
they are equal because they have the
same elements and same
cardinality how about here in number 2
it seems that set a has
more elements than set b but
notice here that 1 was written
twice and 2 was written
twice guys in a set
two to two elements are just the same so
we can just
write these three twos
in a single element likewise with one
so therefore this set a is just equal to
one two three four like in set p
and that's all for this video for more
mad video tutorial please subscribe like
and hit that notification bell
now
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