Venn Diagram and Sets || Mathematics in the Modern World
Summary
TLDRIn this educational video, Ram from Matoklasan explores the concept of sets and their graphical representation through Venn diagrams, introduced by John Venn in 1881. He explains the universal set and how to represent sets within it using circles or geometrical figures. The video delves into subsets and proper subsets, using Venn diagrams to illustrate their relationships. Ram provides examples to clarify these concepts and introduces the power set, explaining its calculation and application. The video concludes with a tip on determining the number of subsets in a set and encourages viewers to subscribe for more informative content.
Takeaways
- 📚 Sets can be visually represented using Venn diagrams, named after John Venn who introduced them in 1881.
- 🎯 The universal set 'U' in a Venn diagram contains all objects under consideration and is depicted by a rectangle.
- 🔵 Circles or other shapes within the rectangle represent individual sets, and points within these shapes symbolize set elements.
- ➡ Venn diagrams are used to illustrate relationships between sets, such as the set of vowels in the English alphabet.
- 🔶 A subset is a set where every element of set A is also in set B, denoted by A ⊆ B.
- 🔷 A proper subset occurs when A is a subset of B, but A is not equal to B, denoted by A ⊂ B.
- 🔄 The concepts of subset and proper subset are analogous to 'less than or equal to' and 'less than', respectively.
- 🔑 Every set is a subset of itself, and the empty set is a subset of any set, which is a fundamental theorem in set theory.
- 💡 The power set of a set contains all possible subsets, including the empty set and the set itself.
- 🔢 The number of subsets in a set with 'n' elements is calculated as 2^n, which includes all combinations from the empty set to the set itself.
Q & A
What is a Venn diagram?
-A Venn diagram is a graphical representation of sets, named after English mathematician John Venn. It uses circles or other geometrical figures to represent sets and often a rectangle to represent the universal set, which contains all objects under consideration.
What is the universal set in a Venn diagram?
-The universal set in a Venn diagram is represented by a rectangle and contains all the objects under consideration. It varies depending on the objects of interest, such as a deck of cards where all 52 cards would be inside the rectangle.
How do you represent the set of vowels in the English alphabet using a Venn diagram?
-To represent the set of vowels in the English alphabet using a Venn diagram, you would draw a circle labeled 'V' inside the universal set rectangle. Inside the circle, you would place the elements 'a', 'e', 'i', 'o', and 'u'.
What is a subset in set theory?
-A set A is considered a subset of set B (denoted as A ⊆ B) if every element of set A is also contained in set B.
What is the difference between a subset and a proper subset?
-A proper subset (denoted as A ⊂ B) is a subset where all elements of A are in B, but A is not equal to B. A subset can be equal to the set it is compared to, whereas a proper subset cannot.
How can you represent a subset relationship in a Venn diagram?
-In a Venn diagram, a subset relationship can be represented by drawing a smaller circle inside a larger one, indicating that all elements of the smaller set are also elements of the larger set.
What is the cardinality of a set, and how does it relate to subsets?
-The cardinality of a set is the number of elements it contains. In relation to subsets, if a set has 'n' elements, its power set has 2^n elements, which includes all possible combinations of those elements, including the empty set and the set itself.
What is a power set, and how is it represented?
-A power set is the set of all possible subsets of another set, including the empty set and the set itself. It is denoted by P(S), where S is the original set. The power set can be represented by listing all the subsets of S.
How many subsets does a set with 'n' elements have?
-A set with 'n' elements has 2^n subsets, including the empty set and the set itself. This is derived from the fact that each element can either be included or excluded from a subset, resulting in 2 choices per element.
What is the power set of the empty set?
-The power set of the empty set contains only the empty set itself, as the empty set is the only subset of itself. Therefore, the power set of the empty set is {∅}.
What is the power set of a set containing an empty set?
-The power set of a set containing an empty set includes the empty set and the set itself. Therefore, the power set is {∅, {∅}}.
Outlines
📚 Introduction to Venn Diagrams and Subsets
This paragraph introduces the concept of sets and their graphical representation through Venn diagrams, named after English mathematician John Venn. The universal set, represented by a rectangle, contains all objects under consideration, such as a deck of cards. Sets are represented within the universal set using circles or geometrical figures. The paragraph explains how to draw a Venn diagram for the set of vowels in the English alphabet. It also delves into the concept of subsets, where if every element of set A is contained in set B, A is a subset of B. The difference between a subset and a proper subset is highlighted, with examples provided to illustrate these concepts.
🔍 Subset Examples and Venn Diagram Application
The second paragraph continues the discussion on subsets with practical examples. It presents a scenario where set A contains numbers 1 through 8, and set B contains even numbers less than ten. The paragraph tests the viewer's understanding with true or false questions about subsets and proper subsets, using Venn diagrams to visually confirm the relationships. It also touches on the concept that every set is a subset of itself and that an empty set is a subset of any set. The paragraph concludes with an exercise to write all possible subsets of a set containing 'r', 'a', and 'm', and introduces the concept of the power set, which is the set of all subsets of a given set.
🌟 Power Sets and Cardinality
The final paragraph focuses on the power set, which is the set of all subsets of another set. It explains how to determine the number of subsets in a set by using the formula two raised to the power of the number of elements in the set, denoted as 2^n. The paragraph provides examples to calculate the power set of a set and discusses the power set of the empty set and a set containing an empty set. It concludes with a tip for calculating the number of subsets and encourages viewers to subscribe for more educational content.
Mindmap
Keywords
💡Venn Diagram
💡Universal Set
💡Subset
💡Proper Subset
💡Power Set
💡Empty Set
💡Cardinality
💡Element
💡Graphical Representation
💡Theorem
Highlights
Introduction to Venn diagrams and their use in representing sets graphically.
Venn diagrams are named after English mathematician John Venn who introduced them in 1881.
The universal set U in Venn diagrams contains all objects under consideration.
The universal set varies depending on the objects of interest.
Circles or geometrical figures inside a rectangle represent sets in Venn diagrams.
Venn diagrams indicate relationships between sets, such as the set of vowels in the English alphabet.
Subsets are sets where every element of one set is contained in another.
Proper subsets are subsets that are not equal to the original set.
Visual representation of subsets in Venn diagrams with a smaller circle inside a larger one.
Examples of determining if a set is a proper subset of another set.
Explanation of the difference between a subset and a proper subset using symbols.
Subsets can be represented as less than or equal to, while proper subsets are like less than.
Examples of determining if sets are subsets or proper subsets with different elements.
Every set is a subset of itself, and an empty set is a subset of any set.
Writing all possible subsets of a set containing specific elements.
Power set is the set of all subsets of another set, denoted by P of S.
The number of subsets in a set with n elements is two raised to the power of n.
The power set of the empty set contains only the empty set itself.
The power set of a set containing an empty set includes the empty set and the set itself.
Transcripts
hello my name is ram and welcome to
another video of matoklasan
before watching this video make sure
that you have watched
my previous video titled introduction to
sets
if not well you can just watch it later
do you know that sets can be represented
graphically using
venn diagram it was named after the
english mathematician
john van who introduced their use in
1881.
in venn diagrams the universal set u
which contains all the objects under
consideration
is represented by a rectangle
note that the universal set varies
depending on which objects
are of interest so if the universal set
is the deck of cards all 52
cards in the deck should be inside this
rectangle
inside the rectangle circles or other
geometrical figures are used to
represent
sets sometimes points are used to
represent the particular
elements of the set venn diagrams are
often used to indicate the relationships
between sets
for example draw a venn diagram
that represents v the set of vowels in
the english alphabet
so the first thing that you need to do
is to draw a rectangle
and draw a circle inside labeled as
v inside the set are the elements
a e i o u which represents
the set of vowels in the english
alphabet
when learning about sets it's very
important to know about subsets
if every element of set a is also
contained
in set b then set a is a subset of set
b in symbol a is
a subset of b notice that if we will
represent
this in a venn diagram set
a here is a small circle inside
set b because all the elements
of set a are also elements of
set b
for example if r
contains the elements 1 3 4 5
6 7 8 9 10 then the possible subsets are
set a and set m set a
is a subset of r because 3 5 and 7
are inside r
m is also a subset of r because 4
8 and 9 are all elements of
set r but how about
proper subset a is a proper subset of b
denoted by this symbol if a is a subset
of b
but a is not equal to b in the previous
example
we can say that r is a subset of
r but
in a proper subset since these are
same sets and they are equal
we cannot use proper subset
so r is not a proper subset of
r so that's the main difference of
subset and proper subset
and use this symbol if you want to refer
to
the proper subset now
let a be the set containing 1 2 3
and b be the set containing 1 2 3 4
determine if a is a proper subset of b
can we say that a is a proper subset of
b
the answer is yes why
because all the elements of a are
in b and a
is not equal to b
when we relate it in inequality
proper subset is like less than and
subset is like
less than or equal to
see
now how about we try these examples
let a be the set containing x such that
x
is a positive integer less than or equal
to a
if we're going to list all the elements
of a we have
1 2 3 4 5 6 7 and 8.
set b has elements x such that x
is a positive even integer less than ten
so listing all those elements we have
two
four six and eight
now true or false
number one is a
a subset of a
this is true because
all elements of a are in a
number 2 is c a subset of
a not is that
c has elements 2 4 6 8
and 10 while a
only has 1 2 3 4
5 6 7 and 8.
well what we need in a
is 10 so since a does not have 10
c is not a subset of
a so this is
false number 3
is b a subset or a proper subset
of c notice here that b
has elements 2 4 6
and 8. c also has these
elements 2 4 6
and 8 and since b is not
equal to c we can say that b
is a proper subset of c
but how about the last term or the last
item a is not
a proper subset of b let's see
a has these elements and b
has this elements now
it's obvious that not all elements of
a are in b you can see here
that b has no one
while a has element one
so therefore a is not really
a proper subset of b therefore
this is true
now how about we use a venn diagram
in this given is b a subset of
universal set u of course because
all the elements of b are inside the
universal set
u the circle b here is inside
u so this is true
how about number two is c
a subset of b
well you can see here that some of the
elements of c
are not in b so
therefore c is not a subset
of b
how about number three a is a subset
of c it's obvious here
that set a
and set c have no common
elements because they don't overlap
so therefore this is also
false a is not a subset
of c
it is also important to note that for
every set
a a is a subset of a
and an empty set is always
a subset of any set
if this is a given set
we can say that the empty set is
a subset of this set containing r
a and m and since we have this
theorem we can say that
this set is a subset
of itself
now let's take a look at this example
write all the possible subsets
of a set containing r a and m
i'll start with the obvious choice the
empty or the null set because according
to the previous theorem
any set has this subset right
then i write the set for each
element
and then i make a combinations of set
with two
elements in this case i'll start with a
and r
then followed by r and m
then followed by the last combination
for the two elements
a and m of course
the last one what we know in the
previous example or in the previous
theorem
that set a is always a subset
of itself so therefore the last subset
is the same set
so here we have eight possible subsets
for the
given and speaking of subsets
power set is the set of all subsets
of another set so the power set of s
is always denoted by p of s
in the previous example we can see here
that if we will write
all these subsets in a single set
then this is the power set
of the set r
a and m
as easy as that so if we want to write
the power set
of this the answer is
this set
now here's my tip if you want to know
the number of subsets
in a given set if a set
has n elements then its power set has
two raised to n
elements now if i will ask you to
find the possible subsets of one two
three and four all you need to do is
to find its cardinality the cardinality
of this
is what one two three four okay it's
four so therefore n
is equal to four using the formula
two raised to four the answer is
sixteen what does this mean
meaning if you want to know the subsets
of 1 2
3 and 4 there are 16
possible sets
how about we try some tricky questions
what is the power set of the empty set
in the previous theorem we all know that
any set has an automatic
subset which is an
empty set so therefore
the power set of this empty set
is just a set containing
itself
how about the second one what is the
power set
of the set containing an empty set
using again the previous theorem we all
know that the empty set
is always a subset of any set
and aside from that this set
also has another subset
which is what yes which is
itself so therefore
we have two elements for our power set
and what are those two elements
yes the first one is the empty set
and the other one is
the set itself
as easy as that
and that's all for this video for more
mad video tutorial please subscribe like
and hit that notification bell
now
you
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