Venn Diagram and Sets || Mathematics in the Modern World

Summary

In 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

00:00

📚 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.

05:01

🔍 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.

10:04

🌟 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.

Keywords

💡Venn Diagram

A Venn Diagram is a type of diagram that shows all possible logical relations between different sets of items. In the context of the video, it is used to graphically represent sets and their relationships. The script mentions that Venn diagrams were named after John Venn, who introduced their use in 1881, and they are particularly useful for illustrating the relationships between sets, such as subsets and proper subsets.

💡Universal Set

The Universal Set, denoted as 'U', is a set that contains all the elements under consideration in a particular problem. In the video, the script explains that the universal set varies depending on the context and provides an example where the universal set is a deck of cards, implying all 52 cards would be represented within the rectangle used to depict the universal set in a Venn diagram.

💡Subset

A subset is a set where every element of one set is also an element of another set. The script defines a subset and uses a symbol to represent it in a Venn diagram, where set A is a subset of set B if all elements of A are contained in B. An example from the script is a set 'r' containing numbers 1 through 10, where a subset could be a set containing only the numbers 3, 5, and 7.

💡Proper Subset

A proper subset is a subset where all elements of one set are contained in another set, but the two sets are not equal. The video script clarifies this concept by stating that if set A is a subset of set B but A is not equal to B, then A is a proper subset of B. It uses the symbol to denote a proper subset and provides an example with sets A and B to illustrate the concept.

💡Power Set

The Power Set of a set is the set of all possible subsets, including the set itself and the empty set. The video script explains that the power set is denoted by P(S) and provides an example where the power set of a set containing 'r', 'a', and 'm' would include all combinations of these letters, as well as the empty set and the set itself, resulting in eight possible subsets.

💡Empty Set

The Empty Set, also known as the null set, is a set that contains no elements. The script mentions that every set has an empty set as a subset and that the power set of the empty set contains only the empty set itself. This is used to demonstrate that even a set with no elements has a power set.

💡Cardinality

Cardinality refers to the number of elements in a set. The video script uses the concept of cardinality to explain how to determine the number of subsets in a set. It states that if a set has 'n' elements, then its power set has 2^n elements, which is a fundamental principle in set theory.

💡Element

An element is a member of a set. The script uses the term 'element' when discussing how to represent sets within a Venn diagram, such as the set of vowels in the English alphabet, which includes the elements 'a', 'e', 'i', 'o', and 'u'.

💡Graphical Representation

Graphical Representation is a method of visually representing data or concepts. In the video, the script explains that sets can be graphically represented using Venn diagrams, which is a form of graphical representation that helps to visualize the relationships between different sets.

💡Theorem

A theorem is a statement that has been proven on the basis of previously established statements, such as other theorems. The script refers to theorems when discussing the properties of subsets and the empty set, using them as foundational knowledge to explain and derive new concepts.

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

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