Basic Mathematics
Basic Mathematics_1.1_Introduction to Sets
Hi, friends. Welcome to the video of Introduction to Sets. In this video, we will discuss about sets and
its two forms. In the end of this video, you will be able to explain set and the two forms of sets. So,
let's define what do you mean by set. Set is defined as well defined collection of object. Let's look at
this example. Here, in this example, the numbers are 1, 3, 5, 7. We are saying that these are the prime
numbers. Means that in this set, we are keeping only those numbers which are prime numbers. We
cannot keep two here, we cannot four here. So, whatever the number will be here, they have a
property that it is either divisible by the number itself or by the number one. So, this well defined
collection. Set is usually denoted by the capital letters. Either by A or B or C or any of the alphabets of
your choice. So, we use to denote the set by the capital letters. And the elements which we are keeping
inside the set, they are usually denoted by the small alphabets. So, a, b, c, x, y z, anything of your
choice. And these elements are written inside the curly brackets. So, the set is denoted by capital
letter A. So, we can say that A is equals to inside the curly brackets a, b, c; a, b, c are the elements of
the set and capital A is the name of the set.
Let's see some more examples for the sets. We said that set is a well defined collection of objects. So,
we can say that odd numbers less than 10. It is a well defined collection of objects. What are the odd
numbers less than 10? It is 1, 3, 7, 9. Then the rivers of India. This is also well defined if we are going
to make a chart for this one, we can figure out the rivers of India, put it in a one curly bracket, we can
say that this is a set. Similarly, the vowels of the English alphabet, a, e, i, o, u.
This can be a well defined set. Let us look what is not well defined and how we can say that this is not
a set. Look. The collection of bright students in the class. This is not well defined in the sense that
there might be a chance that this student is good in one particular activity but he is not good in all the
other particular activity. And the other student can be good in one particular activity and not good in
the other particular activity. So, different students have different capabilities. So, we cannot say that
this student is bright and this student is dull, or this student is dull and this student is bright. So, this
is not well defined. Similarly, collection of renown mathematicians of the world. Different people
consider different mathematicians that he's, yes, he is a very great. Someone will say that, "No, he's
not so great." So, different one has different opinions of the greatness of mathematicians, so this is
also not well defined. Similarly, the collection of beautiful girls. In my opinion, this girl is beautiful. I
will say that, "This is beautiful." But in your opinion, you will say, "No. This is not a beautiful." So, this
is not a well defined criteria, so this collection is also not going to be a well defined collection of the
set.
Now, let's see how we are going to represent the set. Set is usually represented in two forms. First is
Set builder form, other is the Roster or tabular form. So, let's define the Set builder form. What is set
builder form? In this case, all the elements of the set possesses a common single property which is
not possessed by any element outside the set. For example, suppose you have this set a, e, i, o, u.
What is the common property possessed by the set? The common property is that these letters, they
are vowels. Okay? And none of the element in the set is not a consonant.
This means that no letter other than vowel is considered here in the set. So, we can write this set in
the set builder form as like this: V is equals two x, says that x is a vowel in the English alphabet. Now,
let's define the Roster or a tabular form. It says that, it is here in this case, all the elements of the sets
are listed, and they are separated by commas. Means that we are not writing that x says that x is
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possessing this property here. We will especially mention that if it is the vowel, we will write down a,
e, i, o, u, and we will put all those things in the bracket.
Just have a look. Here, the set of all even positive integers less than seven. So, what are the set of all
even positive integers less than seven? It can be only 2, 4 and 6. So, we will mention it there in the
curly bracket, 2, 4, and 6. So, this is called the roster form. Similarly, set of all vowels in the English
alphabet, it is a, e, i, o, u, keep it in the curly bracket separated by the comma. This is the roster form.
Set of all natural numbers which is divisible by 42. If you will find the all the divisors of 42, it is 1, 2, 3,
6, 7, 14 and 21, and the number 42 itself.
You can check it yourself. So, we are putting all these divisors of 42 in one curly bracket separated by
a comma. This is roster form. So, what is roster form? It is actually all the words, letters, or the
numbers written inside the curly bracket separated by a comma. One thing to be remembered here is
that... Look at the first example, it is 2, 4, 6. If we are writing 4, 2, 6, both the sets are same. If we are
writing 6, 2, 4, it is again going to be same. This means that in the roster form, order of the elements
is immaterial.
You can write anything anywhere, but it should be inside the curly bracket.
Key points discussed in this video are sets and two forms of sets. That is Set builder form and Roster
form. How can you use this? Sets are widely used in almost all spheres of life, whether it is home,
office, playground or any other place.
Basic Mathematics_1.2_Empty Sets
Hi, friends. Welcome to the video of Empty Sets. In this video, will discuss about empty sets. At the
end of this video, you will be able to explain empty set. Let's define what you mean by an empty set.
A set which does not contain any element is called empty set. It is also known as null set or void set.
Empty set is usually represented by phi or empty curly brackets like this or by this. So, let's consider
some example of the empty sets. First one, set consisting of all those integers, which satisfies the
equation x^2 + 1 = 0.
So, can you be able find out any integer which satisfies this equation x^2 + 1 = 0. No, it is impossible.
You cannot find any integer, which satisfies the equation, x^2 + 1 = 0. So, this is an example of an
empty set. Let's consider another example. It says that, the set consisting of all those students, which
are presently studying in the class 10th and class 11 simultaneously. So, is it possible that a student in
studying in the class 10th and side by side, he is studying in the class 11th also?
No, answer is no.
So, this set is also going to be in empty set. Next example. Here, this set it consists of all those natural
numbers lying between 1 and 2. If you just remember, the definition of initial number, natural
numbers are the numbers 1, 2, 3, 4, and so on. So, and this set consist of all those natural numbers
which lies between 1 and 2. So, between 1 and 2, there is no natural number. So, this means that, this
is also an example of an empty set.
Key points discussed in this video is empty set and example.
How can you use this? Empty set provides a foundation for building of formal theory of numbers.
Basic Mathematics_1.3_Types of Sets - Part 1
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Hi friends, welcome to the video of Types of Sets Part I. In this video, we will discuss about finite set,
infinite set, equal set, and equivalent set. At the end of this video, you will be able to define finite set,
infinite set, equal set, and equivalent sets. So, let's define all one by one. Let's start with the finite set.
Finite set is defined as a set which is empty or which consist of definite number of elements. Let's
have a example of finite set. Let M be the set of all days of the week, since we know that in a week,
there are only seven days.
So, this means that the set M is finite. Similarly, this set is collection of solution of the equation. X^216=0. So, what is the solution of this equation X^2-16=0? This becomes X^2=16, so X is equals to plus
minus 4. So, this set has only two values, +4 and -4. So, this means that the set is also finite. Now, let's
define what you mean by infinite sets. A set which is not empty and consist of infinite number of
elements is called infinite set.
For example, the set Q of rational numbers is infinite. R, the set of real numbers is infinite, the set of
stars in the universe is infinite. G, the set of all points on the line, then G is infinite. Now, let's define
what you mean by equals sets. Suppose A and B are two sets, then we will say that the sets A and B
are equal, if they have exactly same elements, means, whatever be the elements present in the set
A, those elements are also present in the same set B, then we will say that these two sets are equal.
If that is not the case, then we will say that the sets are not equal or the sets are unequal and A is not
equals to B. Examples of the equal sets are suppose, set A consisting of 1, 2, 3, 4 and B consisting of 3,
1, 4, 2. So, are these sets equal? What is the criteria? That the whatever the elements are present
in the set A, exactly those elements should be present in the set B. So, A consist of 1, 2, 3, 4 and B
consisting of 3, 1, 4, 2. So, obviously, whatever the elements are present in A, they are also present in
B.
We already have discussed that in the sets, the order of the element doesn't matter.
So, in the set B, the order of the elements are changed with the order, but the elements are same. So,
this means that these two sets are equal. Look at the other example, let A be the set of all prime
numbers less than 6. And P be the set consisting of all prime factors of 30. So, what are the prime
numbers less than 6. These are only 2, 3, and 5. And if you will find the factors of 30 especially the
prime factors of 30, it is also the 2, 5, and 3.
So, this means that A and P are equal. Another example is this example where L is consisting of 1, 2,
3, 4 and M is consisting of numbers 1, 2, 3, 8.
So, can we say that these two sets are equal? No. Why? Because the set L consisting of element 4,
and the set M consisting of element 8. So, neither the 4 is contained in the set M, nor the number 8
is contained in the set L. So, this means that these two sets are not equal, they're unequal. Let's define
what you mean by equivalent sets. A and B are two sets, in fact two finite sets, th en we say that A
and B are said to be equivalent, if they have same number of elements. Then, we write A is equivalent
to B.
One thing should always be kept in mind that equal sets are always equivalent, but equivalent sets are
not equal.
We will just discuss the examples of this one also. So, let's consider the examples. Here we have two
sets A and B. A consisting of a, b, c, d, e, and B consisting of 1, 3, 5, 7, 9. So, these two are equivalent
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sets. How they are equivalent? What is the definition? It says that both the sets have same number of
elements. What are the elements we are not bothered about that. We are only bothered about the
number of elements in the both the sets should be equal. So, the set A consisting of five elements
and set B consisting of five elements, so that is why they are equivalent.
But this is not equal. This is one of the example. Because the elements of the set A are a, b, c, d, e, and
the elements of the set B are 1, 3, 5, 7, 9. So, they are not same. So, this means that they are not
equal, but they are equivalent. Another example is the set of the letters of the word "ALLOY" and the
set of the letters in the world "LOYAL". So, if you will look how many words are there in ALLOY, it is
five. And how many words are there in the world LOYAL, it is five.
This means that both are equivalent sets.
Let set A consists of three children of a family, and B consisting of three members of the family. Then,
these two sets are equivalent. Because set A also consisting of three numbers, set B consisting of
three numbers. So, these two sets are equivalent. Key points discussed in t his video are finite set,
infinite set, equal set, and equivalent sets. How can you use this? The concept of these sets will help
you in decision making.
Basic Mathematics_1.4_Types of Sets - Part 2
Hi, friends. Welcome to the video of Types of Sets Part II. In this video, we will discuss about Subsets
and Power Sets. At the end of this video, you will be able to define subsets and power set of a set. So,
let's start with the Subset. If every element of the set A, is also an element of the set B, then A is called
subset of B. Or we can say that A is contained in B, and its representation is A, contained in B. Note
that if at least one element of the set A is not present in the set B, then we say that A is not a subset
of B.
And in the notation form, it is written as A not contained in B. If A is contained in B, means A is the
subset of B, and A is not equals to B. Means all the elements of A, they are present in the set B, but A
is exactly not equals to B, then that set is called proper subset of B. And B is called super subset of A.
For example, look at this one, A = 1, 2, 3, and B consisting of 1, 2, 3, 4. Means that all the elements of
the set A, they are present in the set B, but A is not equals to B because A is consisting of three
elements 1, 2, 3, but B is consisting of four elements 1, 2, 3, 4. So, this means that A is not equals to
B, but A is contained in B. So, this is one of example where A is proper subset of B, and B is called a
super subset of set A. If a set consists of only one element, then that set has a special name called a
singleton set. Few things we should always keep in mind is that, if A is a subset of B, and B is a subset
of A, all the elements of A, they are present in B, and all the elements of B are present in A. The
straight for those shows that both the sets are equal.
That is A = B. And every set is a subset of itself. Every set is contained in itself, so every set is a subset
of itself. That is, if A is a set, A is always contained in itself. So, this is what it is written that A is a
subset of A. And the empty set phi which doesn't have any element. It is subset of every set. In every
set, the empty set is always present there. So, now let's study the examples of subsets. The first one
is the set S, consisting of all students in a school, and the set T consisting of all students in a class. So,
obviously, the set T is smaller than the set S, and all the elements of the set T, they are contained in
the set S. So, T is a subset of S. Second example. It says that set Q, of rational numbers is the subset
of set R of real numbers. Set of real numbers, it is bigger than the set of rational numbers. So, set of
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rational numbers, they are contained in the set of real numbers. Set consisting of all divisors of the 56,
name it as A. And another set B, it consist of all prime divisors of 56. So, obviously, set B is contained
in set A. So, B is a subset of A.
Now, let's define Power Set of a Set. The collection of all subsets of a set A, is called power set of A.
And it is always denoted by P(A). If A is the set, it is denoted by P(A). If the set of consideration is B, it
is denoted by P(B), and so on. In P(A), every element is a set. Now, let's find out a technique. If A has
n number of n or m number of elements, so what would be the number of elements in its power set?
It says that, if A has m number of elements, then the number of elements in its power set is always...
You can always find out with the formula. It is going to be 2^m. Let's consider our first example. Let
A consist of two elements 1, and 2. Then, what would be the power set of the set A? Remember,
power set is set of all subsets of the set A. So, let's first find out all the subsets of the set A. We know
that MD set is a subset of every set, so phi is always there. Singleton 1 is going to be subset of set A.
Singleton 2 is going to be subset of set A, and we know that every set is a subset of itself. So, set A
itself is a subset of A. So, power set of A consisting of four elements. And we can also find it out from
the fact that if A has two elements, then power set of A is going to be 2^2 = 4 elements. So, both the
results are same and these are the elements of the power set of A. In another example, the set S
consisting of three elements x, y, and z. So, by applying the formula, we know that the power set of A
consisting of 2^3 elements = 8. And all the possible power sets, they are given in the figure there.
The third one is, let set S consisting of s, c, h, o, l. This is five letters. They are in the set S. So, power
set consisting of 2^5 elements, means that it has 2^5 subsets. And these are all possible 2^5 subsets
of the set S. Key points discussed in this video are subsets of a set, and power set of a set. How can
you use this? Power set has its application in combinatorial analysis.
Basic Mathematics_1.5_Types of Sets - Part 3
Hi, friends. Welcome to the video of Types of Sets Part III. In this video, we will discuss about Universal
Set and Venn Diagram. At the end of this video, you will be able to define universal set and Venn
diagram. So, let's define first the universal set. Universal set is a set U, which consists of all sets under
consideration as a subset of U. For example, look at this the Venn diagram. The bigger rectangle with
the yellow color is our universal set, and in this universal set, all the subsets will be contained inside
it.
So, the green circle is the subset of the universal set.
So, universal set contains all the sets in itself. Let's look at the example. U is the set of all living things
on the planet. Then, these set of dogs, set of fishes, sets of trees, these are all subsets of the universal
set U. Another example. Suppose, we are considering the human population and we want to study
something about the human population. So, universal set here consists of all people in the world and
different samples at different places, different samples of human population at different places, they
are going to be the subsets of this universal set S. Another set is the set Z of all integers. So, universal
set can be the set Q, which is a rational number or the set R, which is a real number, because set of
rational numbers it contains the set of integers, and set of real numbers also contain set of integers.
So, set of real numbers or set of rational numbers, they can be a universal set for the set of integers.
Now, let's define Venn diagram. Venn diagram is a diagram that shows all possible logical relations
between finite collection of different sets. Means that, if U is a universal set and A and B are two
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subsets of the universal set U, such that is A is the subset of B, or you can say A is contained in B.
Then, how does Venn diagram will look like? Look, the orange bigger rectangle gives you the picture
of universal set. And we know that if U is a universal set, then all the sets they are going to be contained
inside U. So, look at the blue circle and the green circle. Blue circle is the set A and the green circle is
the set B. So, obviously these two sets are going to be contained inside the universal set U. So, they
are inside the orange rectangle U.
Then, there is a condition that A is a subset of B. Means that all the elements of A, they are contained
in the set B. So, the circle A, it is contained in the bigger circle which is the set B. So, this is how we
can show this representation of the set that if U is a universal set in A and B are two subsets of the
universal, such that A is contained in B. So, diagrammatically if we want to show, we can show it with
the help of this Venn diagram. Another example is, let U is the set consisting of 1, 2, 3, 4, so on up to
10. Then, we say that U is the universal set, and we have the subsets of the set U as, A = 2, 4, 6, 8,
and 10. And B = 4 and 6. So, obviously A and B are subsets of the universal set U, and here we can
easily find out that B is contained in A, that is, B is a subset of A. This Venn diagram, it is easily shown
that 4 and 6, which are the elements of the set B, they are actually the elements.
They are also contained in the set A. So, that 4 and 6, they are lying in the overlapping region, and U
is the universal set consisting of all the 10 elements. So, this is how we can represent it with the help
of Venn diagram. Key points discussed in this video are Universal Set and Venn Diagram. How you can
use this? Venn diagrams helps in solving probability problems and understanding logical relations.
Basic Mathematics_1.6_Types of Sets - Part 4
Hi friends, welcome to the video of Types of Sets Part IV. In this video, we will study complement of a
set. At the end of this video, you will be able to define complement of a set. Let's define complement
of a set. Let U be a universal set and A is a subset of the set U, means that all the elements of the set
A, they are contained in the set U. Now, we want to define complement of a set A. So, what is that?
Complement of a set A with respect to the universal set U is the set of all elements of U, which are not
the elements of the set A.
Means that those elements are the elements of the universal set U, but they are not the elements of
the set A, means that the elements of the set U, which are not contained in the element of the set A .
These are called complement of A and the complement of the set A is denoted by A dash, and in the
said builder form, it is written as x such that x belongs to U, but x does not belong to A. Look at this
Venn diagram, here A is a subset of U and A dash is also obviously a subset of U.
But the elements of A and A dash, they are disjoint, means that whatever be the elements in the set
A, they are not contained in the elements of set A dash. So, this is what the definition of complement
of the set A shows. Let's consider some examples.
Example first, here the U set consisting of 10 numbers 1,2,3,4 so on up to 10, and A is a set consisting
of numbers 1,3,5,7,9. We want to find the complement of the set A. So, what is that? Means that the
A dash consist of all those numbers from the set U, which are not contained in A, so A already consists
of 1,3,5,7,9.
This means that in A dash, the numbers 1,3,5,7,9 will not be there, but rest of the numbers will be
there. So, here A dash consists of 2,4,6,8 and 10. Another example where U is the universal set of all
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the students of the class 11 and A is the set of all girl students in the class 11 , so complement of A
would be the set of all boys students of the class 11. So, A dash is the set of all boys in the class 11.
Complement of Universal set is always empty because Universal set is consisting everything, and if
you are taking the complement of a set which contains everything, this means that in the complement
nothing will be there. So, it is always going to be empty. Set and its complements are disjoint. This we
already had seen in the Venn diagram.
Key points discussed in this video is the complement of set. How can you use this? Complement of set
is useful in various programming languages.
Basic Mathematics_1.7_Set Operations - Part 1
Hi, friends. Welcome to the video of Set Operations Part I. In the present video, we will discuss about
union of sets, intersection of sets, disjoint sets and difference of sets. At the end of this video, you will
be able to define union of sets, intersection of sets, disjoint sets, and difference of sets. At the end of
this video, you will be able to define union of sets, intersection of sets, disjoint sets, and difference of
sets. So, let's start with the union of sets. So, let A and B are two sets, then the union of two sets A
and B is another set C, which consists of all those elements, which are either in A or in B.
So, union is always denoted by the symbol 'U.' So, here in the set builder form, we can write it as A
union B consists of all those X such that X belongs to A or X belongs to B. And we can read it as A union
B. Now look at this example; A is equals to 2, 4, 6, 8 and B is equals to 6, 8, 10, 12. So, what is their
union? So, union is going to be 2, 4, 6, 8, 10 and 12. Note one thing that 6, 8 is in A and 6, 8 is in the
set B also, and while writing A union B, we are writing 6, 8 only once.
We are not going to repeat 6, 8 6, 8 again in the set A union B because once the element is present is
equivalent to saying that it is 6, 6, 6 many times. So, whatever be the time it is present in the set A
and B, it will comes only once in the set A union B. This is what it is written here. Another example is
set A consisting of all vowels, that is, a, e, i, o, u and B consists of a, i, u. So, what would be their union?
Because a, i, u is contained in the set A, so their union is the set A itself.
Here, the same thing is written. Now, let's define intersection of sets. Suppose A and B are two sets,
then the intersection of a sets A and B, it is another set C. And it is the set of all those elements, which
belongs to both A and B. Means that the elements, which are common in both A and B, it will come in
the intersection of the sets. If you will look at this Venn diagram. Here, U is the universal set, A is the
green color set and B is your orange color set.
And the intersection portion is that portion of the two circles, which are overlapping.
It means that in the overlapping portion, that overlapping portion it is common both in A and both in
B. So, this is what we are saying that intersection consists of all those elements, which belongs to both
A and B. An intersection is always denoted by the letter down U, you can say. So, symbolically in the
set builder form, A intersection B is set of all those elements X, which belongs to and which belongs
to B. And it is read as A intersection B. Let's define disjoint sets. If A intersection B is equals to phi,
then we say that A and B are disjoint.
How we will say that A and B are disjoint. Because A intersection B consists of those elements, which
are common in A and B. But if there is nothing common in A and B, this needs that both the sets are
individual sets. There is no intersection or no overlapping. This means that these two sets are disjoint
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sets. Look at the figure here. Here A and B, these are two separate circles. So, nothing is common in
between them. So, this means that these are disjoint. Let's look at some examples. Here A is the set
consisting of 2, 4, 6, 8 and B consisting of 6, 8, 10, 12. Then, 6 and 8 are the elements common to both
A and B. So, their intersection comes out to the 6, 8.
Now let's define difference of a sets. Let A and B are two sets. Then, the difference of sets A and B is
the set of elements, which belongs to A, but does not belongs to B. In the set builder form, we will
write it as A minus B is set of all X, such that X belongs to A but X does not belongs to the set B. And
it is read as A difference B, or A minus B. Note one thing that A minus B is not equals to B minus A,
because B minus consists of all those X which belongs to B, but which does not belongs to A.
So, these two differences, they are different. So, this is what it is written here, that the set A minus B
and B minus A are different. So, if we will look at the Venn diagram, U is the universal set, A is subset
of the set U. B is another subset of the set U. Then, A minus B is the dark green portion, and it doesn't
contains the element that portion, which is the portion contained in B also. So, A minus B is the
portion, which is free from the set B. Let's now consider the examples of the difference of the set.
So, let A is equals to a, b, c, d and is equals to b, c, e, f. We want to find A minus B. So, how to do that?
The elements of the set A, which are also present in the set B, we will remove that one. So, A minus
B is a and d because b and c is also present in the B, so we will take it out. So A minus B is a, d. In
another set V is equals to a, e, i, o, u and B is equals to a, i, k, u.
So, what is then V minus B? So, V minus B is e, o because a, i, u is also present in the set B. And what
is B minus K? Remove the elements from the set B, which is present in the set K, so which is a, i, u.
This is present in the set V also, so only element which is not present is V is k. So, this is B minus V. Key
points discussed in this video are: Union of sets, intersection of sets, disjoint sets, and difference of
sets. How can you use this? In this video, we learnt about union, intersection and difference of the
sets. As discussed in the earlier slides, we can see that these sets operations are very much useful in
data analysis, grouping and studying the different characteristics of sets.
Basic Mathematics_1.8_Set Operations - Part 2
Hi, friends. Welcome to the video, Set Operations - Part II. In this video, we will define Cartesian
product. At the end of this video, you will be able to define Cartesian product of sets. So, let's start
with the Cartesian product of set, what you mean by Cartesian product of the set? Let us suppose you
have two sets A and B. The Cartesian product of the set A and B is defined as set of all 'ordered pair'
(a, b) where a belongs to A and b belongs to B, means that the first element should belong to the set
A and the second element should belong to the set B, and it is denoted by A x B.
If we are going to write it in the set builder form, it is written as A x B is equals to the set consisting of
all order pair (a, b) says that a belongs to A and belongs to B. Let us considered the case when the
number of sets are three. So, if A, B, C are the three sets, then the order pair (a, b, c), where a belongs
to A, b belongs to B and c belongs to C is called 'ordered triplet.' And the Cartesian product of these
three sets, it is represented as A x B x C is equals to all those order triplet ABC says that a belongs to
A, b belongs to B, c belongs to C, means first element should belong to the first set, second element
should belong to the second set, and the third element should belongs to the third set. One thing
should be kept in mind that the ordered pair and the ordered triplets are also known as '2-tuple' and
'3-tuples' respectively. Let us consider example, where set A = (a, b, c) and set B = (1, 2). Look at this
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Venn diagram. Here, this oval A is consist of a, b, c and this horizontal oval, it consists of the two
elements 1 and 2.
So, they're Cartesian product is this square consisting of (a, 1), (a, 2), (b, 1), (b, 2), (c, 1), (c, 2). So, this
is all the outcome of the product of A and B. In another example, you have A = (1, 2, 3) and B = (4, 5).
So, let's find out the product. First take 1 from the set A and make an ordered pair with every element
of the set B, so it comes out to be (1, 4), (1, 5). Now, we'll take second element of the set A, and we
will make ordered pair with every element of the set B, so it comes out to be (2, 4), (2, 5).
Similarly, we will do with 3, so it is (3, 4), (3, 5). And B x A, first we will take 4. We will make the ordered
provide all the elements of the set A, so it comes out to be (4, 1), (4, 2), (4, 3). Similarly, with 5, it is (5,
1), (5, 2), (5, 3). Note that, (1, 4) belongs to A x B and (1, 4) does not belongs to B x A. So, the shows
that A x B is not always equals to B x A. Key points discussed in this video is Cartesian product of sets.
How can you use this? In this video, we learned about Cartesian product of sets. Cartesian product
gives us the concept of dimensions.
For example, Cartesian product of two sets gives us the concept of two-dimensional space, Cartesian
product of three sets gives us the concepts of three-dimensional space. Similarly, Cartesian product
of N sets takes us to N-dimensional space. Hence, it helps us to locate the object in space and thus,
becomes very important part in study.
Basic Mathematics_1.9_Important Theorems on Sets
Hi friends. Welcome to the video of Important Theorems on Sets. In this video, we will discuss about
different theorems on union and intersections. At the end of this video, you will be able to use
different theorems on union and intersection of sets. So, let's start, first with the theorem on union of
two sets. So, let A and B are two finite sets such that their intersection is empty. Then, we can define
number of elements in A union B is equals to number of elements in A plus number of elements in B.
And if A and B are two finite sets such that intersection is non-empty, then in this case, the theorem
comes out to be n of A union B is equals to n of A plus n of B minus n of A intersection B. Let's discuss
the case when there are three sets. So, let A, B, C are three finite sets with the condition A intersection
B not equals to Phi, B intersection C not equals to Phi, A intersection C not equals to Phi, and A
intersection B intersection C not equals to Phi. Then what will happen? It comes out to be n of A union
B union C is equals to n of A plus n of B n of C minus n of A intersection B minus n of B intersection C
minus n of A intersection C plus n of A intersection B intersection C.
Now, we will study the DeMorgan's Law. It says that if A and B are two sets, then A union B all
compliment is equals to A compliment intersection B compliment, and A intersection B all compliment
is equals to A compliment union B compliment . And if A, B, C are three sets, then in this case it is
defined like this; A union B union C all compliment is equals no A compliment intersection B all
compliments intersection C compliment. Similarly, for in the intersection case, it comes out to be
union of their compliments. And for n number of sets, it is defined like this. Now, the Commutative
Law: If A and B are two sets, then A union B is equivalent to B union A, and A intersection B is equivalent
to B intersection A. Associative Law: It says that if you have three sets A, B, and C, then first take A,
and in the bracket take the union of (B union C). So, (A union B union C is equals to, first taking the
union of these two sets A and B, and then take the union of the set C. So, A union B union C is
equivalent to, first take, in the bracket put A union B, and then take it with the set C. Similarly, for the
intersection, A intersection B intersection C is equivalent to, first take (A intersection B), and then,
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keep it with the intersection with C. Distributive Law: It says that for the sets A, B, and C, the following
law holds. A union B intersection C is equals to A union B intersection A union C, and A intersection B
union C equals to A intersection B union A intersection C.
So, let's consider this example. It says that, in a school there are 20 teachers who teach mathematics
or physics. Of these, 12 teaches mathematics and 4 teaches physics and mathematics. So, how many
teaches physics? So, let's visualize it with the help of Venn diagram, and then try to solve it by making
use of the previous theorems. Look here. Total number of teachers in a school is 20, of which 12
teaches mathematics and 4 teaches physics. So, we have to find out how many teaches physics. Look
here. M is the set which consist of the teacher who teaches mathematics. P is the set, consist of
teacher who teaches physics. And it is given that n of M union P is equals to 20, and n of M is equals
to 12, and n of M intersection P is equals to 4, maths and physics. So, it is intersection which is equals
to 4. So, we have the formula, n of M union P is equals to n of M plus n of P minus n M of intersection
P. Whatever is given in our question we will put it there, and what is not given we will calculate it. So,
let's see here. So, we have to find n of P. So, it comes out to be this thing. n of P is equals to n of M
union P minus n of M plus n of M intersection P, which is equals to 12. Key points discussed in this
video are; different theorems on union and intersection of sets. How can you use this? In this video,
we discuss about different laws which are used in writing programs and expedite the work. With the
help of these laws, complicated problems can be solved with very less effort.
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