1. Introduction to Classical Sets or Crisp Sets in Fuzzy Set Fuzzy Logic by Mahesh Huddar
Summary
TLDRThis educational video explores the concept of classical or crisp sets, which are collections of distinct objects. It explains their mathematical representation through roster or tabular form and set builder notation. The video also covers various types of sets, including finite, infinite, empty, subsets, proper subsets, universal sets, single sets, equal sets, equivalent sets, overlapping sets, and disjoint sets. Each type is illustrated with examples to clarify their unique characteristics and differences.
Takeaways
- đ A classical set, also known as a C set, is a collection of distinct objects, such as all positive integers or all planets in the Solar System.
- đą The mathematical representation of classical sets can be done in two ways: roster (tabular) form and set-builder notation.
- đ In roster form, elements of a set are listed within curly braces, like {a, e, i, o, u} for vowels.
- đ Set-builder notation uses a variable and conditions to describe the set, for example, {x | x is an odd number < 10}.
- đ Types of classical sets include finite sets with a limited number of elements, infinite sets with unlimited elements, and the empty set with no elements.
- đ A subset is a set where every element is also in another set, denoted by X â Y.
- đŻ A proper subset has all elements of the original set but also includes at least one additional element not in the original set.
- đ The universal set contains all elements under consideration in a given context, like all animals on Earth.
- đ Equal sets have the same elements, regardless of order, while equivalent sets have the same number of elements but can be different sets.
- âïž Disjoint sets are sets that have no elements in common, whereas overlapping sets share at least one element.
Q & A
What is a classical set?
-A classical set is a collection of distinct objects, such as a set of all positive integers or a set of all planets in the Solar System.
How are classical sets mathematically represented?
-Classical sets can be represented in two main ways: roster form (listing all elements within curly braces) and set-builder notation (describing the elements that satisfy certain conditions).
What is an example of a classical set in roster form?
-An example of a classical set in roster form is the set of all vowels in the English alphabet, represented as {a, e, i, o, u}.
How is the set of odd numbers less than 10 represented in set-builder notation?
-The set of odd numbers less than 10 is represented in set-builder notation as {x | x is an odd number and 1 †x < 10}.
What is a finite set?
-A finite set is a set that contains a limited or countable number of elements, such as the set of odd numbers less than 10.
Can you explain what an infinite set is?
-An infinite set is a set that contains an unlimited or uncountable number of elements, like the set of all integers greater than 10.
What is an empty set or null set?
-An empty set or null set is a set that contains no elements at all, such as the set of all integers greater than 7 and less than 8.
How is a subset defined?
-A set X is a subset of another set Y (denoted as X â Y) if every element of X is also an element of Y.
What is the difference between a subset and a proper subset?
-A proper subset is a subset where the cardinality of the subset is less than the cardinality of the original set, meaning the original set contains at least one element not present in the subset.
What is a universal set?
-A universal set is a set that contains all the elements under consideration in a particular context or application, such as a set containing all animals on Earth.
How are equal sets defined?
-Equal sets are sets that contain the same elements, regardless of the order of those elements. For example, {1, 2, 6} and {6, 1, 2} are equal sets.
What is an overlapping set?
-Overlapping sets are sets that share at least one common element. For instance, if set A = {2, 6} and set B = {6, 12, 42}, then A and B are overlapping sets because they both contain the element 6.
How are disjoint sets characterized?
-Disjoint sets are sets that do not share any common elements. For example, if set A = {1, 2, 6} and set B = {7, 9, 14}, then A and B are disjoint sets.
Outlines
đ Introduction to Classical Sets
This paragraph introduces the concept of classical sets, which are collections of distinct objects. Examples provided include sets of positive integers, planets in the Solar System, states in India, and lowercase letters of the alphabet. The mathematical representation of classical sets is discussed, highlighting two main methods: roster (tabular) form and set-builder notation. Roster form lists all elements within curly braces, while set-builder notation describes the elements using a variable and conditions. The paragraph also explains different types of classical sets, such as finite sets with a limited number of elements, infinite sets with an unlimited number, and the empty set with no elements.
đ Exploring Subsets and Other Set Types
This paragraph delves into the concept of subsets and proper subsets, explaining that a subset is a set where every element is also found in another set (Y), while a proper subset contains all elements of the original set (X) but also has at least one additional element not in X. The paragraph also discusses the universal set, which contains all elements under consideration in a given context. It introduces the single set or unit set, which contains only one element, and equal sets, which have the same elements regardless of order. The concept of equivalent sets, which have the same number of elements, is also explained. The paragraph concludes by differentiating between overlapping sets, which share at least one common element, and disjoint sets, which have no common elements. The speaker encourages viewers to like, share, and subscribe for more educational content.
Mindmap
Keywords
đĄClassical Set
đĄRoster or Tabular Form
đĄSet Builder Notation
đĄFinite Set
đĄInfinite Set
đĄEmpty Set
đĄSubset
đĄProper Subset
đĄUniversal Set
đĄSingle Set or Unit Set
đĄEqual Set
đĄEquivalent Set
đĄOverlapping Set
đĄDisjoint Sets
Highlights
Classical sets are collections of distinct objects, such as all positive integers or all planets in the Solar System.
Mathematical representation of classical sets can be done in roster or tabular form, listing all elements.
Set builder notation is another way to represent classical sets, using conditions to define elements.
Finite sets contain a limited number of elements, such as odd numbers less than 10.
Infinite sets have an unlimited number of elements, like all integers greater than 10.
Empty sets, or null sets, contain no elements, such as numbers greater than 7 and less than 8.
A subset is a set where every element of one set is also an element of another.
A proper subset is a subset where the original set has at least one element not in the subset.
The universal set contains all elements under consideration in a particular context.
A single set, or unit set, contains only one element.
Equal sets have the same elements, regardless of the order in which they are listed.
Equivalent sets have the same number of elements, even if the elements themselves are different.
Overlapping sets share at least one common element.
Disjoint sets have no elements in common.
Understanding classical sets is fundamental for various mathematical concepts and applications.
Transcripts
welcome back in this video I will
discuss what are classical or risp sets
what is the mathematical representation
as well as what are the different types
of C sets with simple
examples classical set is a collection
of distinct objects for example a set of
all positive integers it is known as one
classical set because it contains a
distinct positive integers here
similarly a set of all planets in the
Solar System a set of all states in
India a set of all lowercase letters of
alphabet all these are an examples of
classical sets now coming back to the
mathematical representation of classical
or C sets there are mainly two ways to
represent the classical sets the first
one is known as rooster or tabular form
in rooster or tabular form we use to
represent the set with a simple alphabet
let us say that in this case a is a set
which is equalent to within clyra we
used to write all the elements present
in this set here so in this case a is
equalent to within carbras I have
written a e i o u the meaning is set a
is the collection of all ows in this
case similarly I have written one more
example set of odd uh numbers less than
10 uh which is represented as B here B
is equal to within curly brace I have
listed all the elements uh in this case
the number which should be less than 10
and they should be odd here they are 1 3
5 7 9 in this case so this is another
example which is represented in the form
of rooster or tular form here the second
one is set builder notation the same two
examples were represented in another way
uh in this case a IOU uh it is
represented something like this a is a
set name which is equalent to within
clyra I have written X here the meaning
is uh the set a contains all those
elements X such that X is a all in
English alphabet so we have written this
condition that is such that if x is a o
then it is present in a here if it is
not a o it is not present here so that
is what the meaning of this
representation here I will take the
second example that is set containing
odd numbers less than 10 which is
represented as B here B is equal to it
contains X that is it contains X if or
such that if x is greater than or equal
to 1 and less than 10 here it should be
greater than equal to one and it should
be less than 10 here that is it should
be less than 10 that is a condition here
and another condition is what it should
be odd number how can you write it it
should be greater than equalent to one
and less than 10 along with that X mod 2
should be equalent should not be
equalent to zero if it is equalent to Z
it will become even number it is not
equalent to zero it will become a odd
number for example if x X is equal to 1
1 is in this particular range as well as
1 mod 2 is equalent to 1 the meaning is
it is not equalent to zero it is present
in V here similarly if I take two here
two is in this range 2 more 2 is
equalent to zero because it is
equivalent to zero uh it is not present
here if it is not equivalent to zero
then only it will be present in this
particular set here let's take another
example three three is in in this range
3 m 2 is equal to 1 one is not equalent
to zero so it will be present in this
set here and so on next we will discuss
what are the different types of
classical or risp sets here the first
one is finite set the meaning of this
one is a set containing finite number of
elements for example if you say that s
is equal to it contains all those
elements x if x is in the range of 1 to
10 and X mod 2 is equalent to is not
equalent to zero the meaning is all odd
numbers less than 10 here so in this
case we have 1 3 5 7 9 uh they are the
finite number of elements present in the
set so it's called as finite set here
coming back to the second one infinite
set if a particular set contains
infinite number of elements then it is
called as infinite set in this case s is
equal to X such that X is a integer and
X is greater than 10 here that is
nothing but X contains 11 12 13 14 till
Infinity all those elements are present
in this set hence it is called as
infinite set here the next one is empty
set or null Set uh the meaning of this
one is a set containing zero number of
elements or there are no elements in a
particular set then it is called as a
null set or empty set denoted by a
special symbol called as five here s is
equal to contains all those elements x x
is a integer X is greater than seven and
less than 8 there is no number which is
greater than 7even and less than 8 here
hence it contains zero number of
elements that's reason it is called as
empty set here
coming back to the next example that is
a subset here a particular set X is said
to be a subset of Y denoted by X subset
y if every element of X is an element of
Y here for example if you say that Y is
equal to 1 2 3 4 5 6 and X is equal to 1
2 x is called as a subset of Y because
all the elements present in X are also
present in y also hence it is called as
a subset here
coming back to the next type of set that
is called as a proper subset the term
proper subset can be defined with a
simple line that is it is a subset but
not equal to so we will try to elaborate
it let's say that X is a set and it is
called as a proper subset of Y denoted
by something like this if every element
of X is an element of Y and the very
important condition is the cardinality
of X should be be less than the
cardinality of Y the meaning is y should
contains at least one more element which
is not present in X here for example if
you take this one y is equal to 1 2 3 4
5 6 x is equal to 1 2 all the elements
present in X are also present in y me X
is a subset as well as X is a proper
subset because uh y contains at least
one element which is not present in X
here so that's the reason it is a proper
subset here for example Y is equal to we
have 1 and 2 and X is equal to again 1
and 2 so if you look at here x is a
subset of Y because uh X is a subset of
Y because all the elements present in X
are also present in y here but X is not
a proper subset of Y because all the
elements of X present in y that's true
but y should contain at least one
element which is not present in X here
because it does not contain such element
X is not a proper subset of Y in this
case I hope you understood the
difference between subset and proper
subset coming back to the next one there
is something called as the universal set
Universal set contains all the elements
in a particular context or application
for example uh let's assume that U is a
uh Universal set containing all animals
on the Earth so with respect to animals
all are animals are present in this
particular set hence it is called as a
universal set here the next one is
single set or unit set if a particular
set contains single element or only one
element it is known as a single t or
unit Set uh I have given an example here
s is equal to it contains x x is a
integer and X is greater than 7 and less
than n there is only one possibility
that is eight hence it is called as
singl T set here coming back to the next
one that is equal set if two sets
contains the same number of elements the
order may be different but the same
element should be present then they are
called as equal sets if you see here a
is a set containing three elements 1 2
Six B is a set containing three elements
612 both of them are containing same
elements but the order is different
there is no issue with respect to order
if the elements are same then the two
sets are called as equal sets here
coming back to the next type that is
equivalent set two sets are said to be
equivalent if the number of elements are
same for example you can see here a is
equal to 1 to 6 there are three elements
are there b is equal to 16 17 22 again
three elements are there the number of
elements in a are equalent to number of
elements in B hence they are called as
equivalent sets here they are not equal
sets they are equivalent sets in this
case the next one is overlapping set a
particular set two sets are said to be
overlapping if there is at least one
common element if you look at here a is
equal to 26 B is equal 6 12 or 42 here
six is present in both the sets hence
they are called as overlapping sets here
uh the last one is disjoint sets two
sets A and B are called as disjoint if
they do not contain even one common
element here so if you look at this
example a is equal to 126 B is equal to
7914 a and b does not contain a single
common element hence they are called as
disent setes Industries I hope the
concept of uh classical sets is clear if
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