So, this is a course on algebra, the number is 203B. And in this course, we will essentially

introduce the algebra again to you. I am sure you have all done algebra in some form of

other, used algebra in many forms actually. But, what I am going to talk about is the

modern version of algebra, the way it is used to understand and study different mathematical

concepts. And that is something that perhaps many of you have not seen, maybe partly seen,

but not seen as a development from the basics to the higher level and that is what we will

do in this course. The focus will be on abstractions that how

do we look at various concepts, abstract out the properties of those concepts, and then

study properties in the abstract level. What would be the advantage? Let me just write

down. The advantages of abstraction, first advantage of abstraction is that it allows

us to, that is very clear that it allows us to study concept without unnecessary details.

It gets rid of the specific details that are not relevant, and you can just look at it

only at the core aspects. And the second one also it is extremely important

is that it allows

us to study common properties of different concepts. You have multiple concepts, you

abstract their properties out, some of there will be common. When you once you study the

common property that is applicable across the concepts.

We will see the examples of these, I am sure you have come across examples earlier as well.

And third one, which is again very important, is that it allows us to take out the properties

that you have studied in abstract, and apply it into a completely different context which

is another way of saying that it allows for generalizations. So, we there is a specific

concept we abstracted the properties out of, study those property and now this is the abstract

world, so we can apply it anywhere that we wish to.

So, in a sense what we study would be much more general version of what that specific

thing we started with from. That’s three very big advantages of abstraction. Of course,

there is down side of it also, which is that it becomes perhaps a little boring or rather

too abstract too mathematical, i.e the intuition gets lost many times and that you see them

you know lot of symbols and manipulation of those symbol and it is not clear why exactly

are we doing it. So, it is therefore, it is very important to keep in mind the concrete

concepts from which we have abstracted out, so that we never lose sight of what is the

effect of our abstract manipulations on those concrete concepts. Any questions?

Well, let us see some example abstractions and these are something that you have already,

come across. What are the two most fundamental objects in mathematics? Numbers, yes numbers

are one, clearly. Operators of that’s that you are talking of the that’s an abstraction

of when you want to study certain operations and then you use those, but I am saying not

yet going into the abstract world in the concrete setting in mathematics, what are the two most

fundamental objects it will we like to study. One is numbers. Second?

Properties. Properties of various things sure but objects,

for concrete objects that you want to study. You have studied it for long in school.

Variables. Variables is again an abstraction, use variables

to study some concrete things. Well, geometric objects, curves, planes and so various surfaces

that’s also come from the real life, just like numbers in the sense come from real life

by count, you want to counts certain things. So, you introduce and use new numbers. Similarly,

in real life the whole world is you know lots just the whole collection of geometric objects

and you would like to study how these geometric objects behave, what are their properties.

So, with this in mind, that is how the mathematics starts in the ancient times; the study was

done for numbers or about numbers and about the geometric objects. And already the abstraction

particularly in the case of geometric objects was found extremely you useful. Take for example,

a circle that’s a geometric object, now if I want to study properties of a circle;

one way is to draw the circle and see now geometrically various properties by drawing

whatever other curves and intersections. Do you know another way? Something that you have

been using in the school very. Yes, so use compass, that’s geometric way

of studying circle and you draw circles, lines and then see how they interact. Learning geometry,

yes, you been that’s what you did in the school. The all this basic curves you studied

using coordinate geometry and what is coordinate geometry give is, for a circle instead of

drawing a circle, we write this equation x square plus y square equal to r square and

this represents a circle. So now, we introduce coordinates here which is center is at point

zero zero and there are any point of the circle is at coordinates are x y. And, the radius

of the circle is r and then this circle is represented by this equation x square plus

y square is equal to r square. And then suddenly we can study the circle

by using this equation. You can differentiate it, get the tangent, equation of the tangent

at various points, you can look at multiple two circles and see whether they intersect.

You can do all that you would do by drawing it pictorially, you can do it algebraically.

Now this is already an abstraction, because on its own, this equation is just a collection

of symbols. There is an x, there is a two on top of x, then there is a plus sign, then

there is y, two on top of y and then etc. What we do is we assign meaning to this and

the meaning we assign is that the x represents the x coordinate value of a point on circle,

the y represents the corresponding y coordinate of that point and in this is a square x with

two on top is again a representation of multiplying x with x itself and so on.

So, what we did was that we use this abstraction to represent the circle with an equation and

in such a way that all the properties of that circle are inherent in this equation and then

we study this equation. And then you as you would recall that coordinate geometry is extremely

useful in deriving properties of geometric objects. Things that you would not be able

to prove using just the pictures. You can prove using this abstraction. Okay, can you

remember or give me an example of a property that you can, some non-trivial property that

you could prove using coordinate geometry. Tangent to a hyperbola does not intersect

what. The curve

The curve ok, I will believe you, I do not recall this property, but he saying, so that

tangent to hyperbola does not intersect the curve at any other point in it.

Yeah, there are two parts of the hyperbola, and they it doesn’t intersect. Okay fine.

That you can prove using this. But if you were to prove it using geometry, how would

you do that? You could draw a the hyperbola and draw a tangent at one point and observe

that it does not intersect, but that doesn’t mean that for no hyperbola for any point on

the hyperbola the tangent will not intersect the curve. So, not only you get an easier

way of studying property you get a more powerful way of studying properties by applying this

abstraction. So, that’s the observation that we can prove. So, that is very important

observation and that already demonstrates this power of abstraction.

Now, later on in this course, I will show you further abstraction of the same. There

is you have a circle we abstracted it out as equation and then I will go one more step.

This equation and is study of this properties comes under coordinate geometry. I will go

one more step and abstract data out even more and that that field is called algebraic geometry.

So, we will use more algebra to abstract out the circle and similarly other curves and

use that representation to study properties and it turns out that that representation

is even more powerful than the coordinate geometry abstraction. And we can prove further

or even more properties using that abstraction than with coordinate geometry.

Okay, let’s stay on this picture for little more while, here you see that equation x square

plus y square is equal to r square. This seems like an equation over numbers. Now, there

is addition, there is multiplication in here and addition and multiplication is what we

do for numbers. So, that is a curious things that we have that exists here. Firstly, there

is this geometric object that is been seemingly translated into numbers. In fact that’s

what the name coordinate geometry represents that you assign coordinates and coordinates

are nothing, but collection of numbers to every point and therefore represent a geometric

object as a collection of points and or collection of numbers and then you see properties or

relationship between those numbers that is given by this equation.

Second curious thing about this is that not everything in this is a number. In fact, almost

nothing is a number x is not a number, y is not a number, r is not a number. Numbers are

well; one, two, three, four, one point one, if you want to allow reals, square root two.

Those are numbers. Is x a number? y? r? These are symbols they are not numbers. So, how

can we treat them as numbers. We are seen to be treating them as numbers here. We are

multiply x to itself, adding it to y r y square. So, how can we treat them as a number? Is

something not right with this or are we taking too many liberties with our notations?

There is some abstraction. There is an abstraction that is happening

here is yes, but we need to be very careful when we abstract in precisely defining the

meaning of every symbol that we use in the abstraction. So, when we say that x, y and

r, here are going to be thought of as numbers, is that justified?

Set of numbers. If any one of, so we are at liberty when we

are doing abstraction we are at liberty to assign a meaning to any symbol that we introduce.

What we need to ensure is that that meaning is consistent with the operations we carry

out. So, if you say x represents say set of numbers, collection of numbers, then what

does this x square represents? In what way the collection of numbers of x square related

to collection of number of x. And we say that this collection is multiply

every pick number in from one collection to every number in other collection and then

that overall number is an x are they collected is that what we say. That is not what we intend

to mean anyways. Remember our intention is there is some num x y represents, specifically

one coordinate point. It represents many coordinate point but take one coordinate point then that’s

a number and then you take a square of that number in x square and y square takes the

square over y coordinate. The thing is that x and y don’t represent

a single number. They don’t stand for a single number, they stand for a collection

number that is correct. Right? But that collection of number is also not any collection. x represents

the numbers that correspond to x coordinates of points lying on the circle. So, what is

the meaning that we assign to x therefore here? Do we say that, either we can say just

draw the circle look at the x coordinates of all the points and those represent those

are represented by x here? But that’s really a kind of messy, firstly somebody has to draw

the circle then assign the meaning or atleast imagine a circle in a mind. Instead what we

can say here is it x represents any number, y represents any number such that collectively

they satisfy this equation, that x square plus y square is equal to r square. Does that

make sense? So, x is some number here, y is some other

number here. They need to satisfy this relationship x square plus y square is equal to r square,

r is a fixed number. r is not does not change. So, we can take r let’s say one itself.

Then we say this equation corresponds to collection of all pairs of numbers, which satisfy this

- for x if you substitute the first of the pair, for y substitute second of the pair,

then this equation must be true. So that’s the interpretation we assign to this abstraction.

But again it is not yet very let’s say precise, because I said x can be in any number and

y can be any number and such that they together satisfy this equation.

But when I say number what do I mean? Do I mean integers? Do I mean rational numbers?

Do I mean real numbers or do I mean complex numbers? There are different types of numbers

also. What would you suggest we should treat x as? What kind of number? Real numbers? Yeah,

that’s a natural way of thinking, because this geometric object this circle we are thinking

over a real plane. So, we can therefore, think of x and y as a real numbers. That’s good.

But suppose I want to study the rational numbers lying on the circle, that is points with have

rational coordinates, which lie on the circle. Then we can say that the it is all the rational

numbers x and y, which satisfy this equation are the points of interest to us, which mean

that we are saying that now no longer think of x and y as real numbers, but think of them

as a rational numbers. The equation remains the same just that we are saying assigning

a different meaning to x and y and we can change the meaning maybe some other time we

want to study all complex number satisfying this equation, so we change the meaning to

complex numbers or something else. So, that abstraction remains the same, but the meaning

we assign to it changes, okay, which is very good thing, because then if we study the properties

of this in abstract domain, then no matter what meaning we assign to this eventually,

those properties will continue to be true, fine. So that that is a important observation.

So, we may write it down, this is not hundred percent true because each meaning will have

its own quirks also for at least the common properties we can handle simultaneously. So,

its means we would like ideally to study this equation without assigning meaning to x and

y that’s what you want to do right. I have at least hopefully managed to convince you

that that’s a good idea. Let’s not think of x is a rational number, let us not think

of it as real number. That we will do later when we require, as and when required. To

begin with in a real abstract sense I will not assign any meaning to x and y. Are you

with me? Do you agree with me that it is a good idea? Because it is a dangerous statement

that I making. Because if you do not assign a meaning to x and y, what is it meaning of

x times x, how can you add x and y? .

That’s one way of doing it, but then you have to do it at least three examples that

I give rational, reals and complex. For each of them you have to then study it separately.

. Yes what, but what if. So, each addition may

have slightly different properties. So, then you will have to worry about those peculiarities

of each meaning. Ideally in abstraction we will would like to get rid of the specific

quirks of the particular use and you just like abstract it out so that we can study

objects, which are applicable across multiple meanings, which in this case we would be able

to do if we don’t assign meaning to x and y. The only hitch there is that and it is

a big hitch that then how do we say how do we define addition and multiplication for

x, for y for a combination of these. Let’s assume some properties of addition.

About. Addition.

Addition, yes of course that is a solution, yes, you have given it.

So, let’s attack in a more direct fashion, the concept of numbers itself. What is a number?

Okay, who can tell what is a number is? Any object that is countable.

Any object that is countable that is a number, but there are uncountable reals are uncountable,

so that is not a good enough definition. .

. .

By countable? .

Either power set of count countable . .

Okay so, let’s may give another argument against this. Let’s pick up a collection

of countable objects, which is maybe collection of all stars in the galaxy. Are they numbers?

Yes countable, surely. Are they numbers? Can you add two stars and get a third star? How

about multiplying two stars? That’s not a good definition. Let’s try to cover with

a better definition of a number. It is an ordered set with addition and multiplication.

That is a good start; yes, that is a good start. See what we really want to do with

numbers is arithmetic on it. We have four fundamental arithmetic operations addition,

subtraction, multiplication and division. We would like with that’s essentially how

we play around a number right. You do these operations for numbers. Now of course for

certain collection of numbers not all four of these operations are possible. For example,

for integers, division is not possible. Addition, subtraction, multiplication is possible. But

some others for example, rational numbers division is also possible. So, leaving a side

this slight distinction we would like to do arithmetic with numbers.

So, let us define numbers as the collection of objects on which we can do arithmetic.

But then we have to now define what does it mean to do arithmetic and you pointed already

one important property how doing arithmetic the closure; that is if I add two objects

I should get a third object from my collection. Now, we are dealing with the numbers has just

as object we are not assigning any other meaning to them. So, we just have we now therefore,

have to say what it means to do arithmetic on those objects. So, let us try to write

down the properties we want of those objects.

So, this is a formal x definition of the closure property, that capital A is the collection

of objects that we are going to think of as numbers. The first requirement is that this

collection of objects must admit the addition operation, which I am going to represent with

the symbol plus. This satisfies the following properties the first one is the closure property

that for any a and b in the collection a plus b must be defined and must be a another object

in the collection. Okay? What other property should addition have?

Associativity. Associativity, excellent, that for every a,

b, c you want to add all three of them. So, it does not matter in which order you add

them. Anything else? Commutativity yes, a plus b is same as b plus a. There are more

properties that are required. That are required means that does they do hold for typical numbers

that we have in mind. So, we should abstract those properties out as well. One very important

of these properties is this very special number 0. It has a property that if you add 0 to

any other number, the number does not change and the 0 is called the identity of addition.

So, when you add 0 to a, you get a itself and finally, that is negative numbers or this

property allows you to define subtraction. Just addition is not the sufficient to operate

a numbers, you will also like to do subtraction. Subtraction is simply when you say a minus

b we are adding a with minus b and what is minus b? Minus b is that that number, which

you when add to b you get 0. So, the fact that minus b exist is guaranteed by 5th property.

So, we need this property in order to define abstraction in order to define subtraction.

And in terms of abstraction, we call this property the inverse property the existence

of inverse of a every element. So, these are the 5 properties we require in order to define

addition and subtraction. Have I missed out something, then your rather can you think

of some other property that addition and subtraction should satisfy? So happens that this is the

comprehensive set of properties that the addition operation satisfies for numbers.

And now with this written down, we can define numbers to be any collection with an addition

operation subtraction comes for free which satisfies all these 5 properties. Of course,

this is not quite there because a numbers also have multiplication and division and

I am not using that. So, we should also have division and multiplication definition. But

for now let us just stay on these properties and I will introduce the multiplication, division

slightly later to have the numbers. And the reason why I am stopping here for now is that

this properties for a collection is already a very interesting abstraction.

And, I am sure you probably have seen this, that this collection So, this collection with

the addition operation defined as we just did is called a commutative group. The name

has its own the history. Let’s not get into that why is it called a group, but the key

thing or important thing is that groups are an abstraction, which are extremely useful.

We started with numbers, in order to abstract that we to came to groups. But like I initially

had suggested that once you do an abstraction, you can apply to completely different domains

and that is true with groups as well. Let me give you an example of course, the

obvious examples of groups you already know. Integers with addition operation that is a

group. That’s how we started actually. Not just integers, you take and take rational

with addition, reals with addition, complex numbers with addition. These are all groups

for obvious reasons commutative group that is. By the way I should have said that that

if you drop property three which is the commutativity property. In case of numbers we don’t need

to drop it that this is definitely satisfied by numbers.

But since we have would like to generalize this notion and apply it in other domains

also there are situation where this property doesn’t exist. And in that case, we will

do away with this property for those situations and if you drop property 3 and the remaining

property satisfied by a collection, in that case the collection is called simply a group.

So now, come back to some examples. So, these are the obvious examples coming from numbers.

Let me give you an examples for groups, which do not come from numbers. Okay, any of you

have a suggestion. Have you come across groups some time earlier?

Matrices. Matrices, the excellent example; yes, so let’s

say n cross n matrices under addition that’s a group, we can the identity element is the

all 0 matrix. You can add. It is a commutative group also we can add the all use the other

properties are easy to see that they are true. Okay. More examples.

Permutation. Permutation, yes, collection of the permutations;

let’s say look at the permutations on numbers 1 to n. Each permutation by definition is

a mapping from one to n to itself, which is a one-one onto mapping. So, what is the addition

operation for the permutations? How do you add two permutations?

Compose. Compose, under composition. So, composition

of two permutations is a permutation. So, let’s just quickly go through all the property.

Closure is true. Associativity is true again it is very easy to see. It is not commutative.

If we take two permutations, and it matters in which order you apply those permutation.

phi 1 composite phi 2 is not necessarily equal to phi 2 compose with phi 1. phi 1 may map

1 to 2, phi 2 may map 2 to 3. So, if you apply first phi 1 and then phi 2 then you will map

1 to 3. On the other hand, if you apply phi 2 – first,

phi 2 may map 1 to 7; and phi 1 may map 7 to 20, if you apply first phi 2 and then phi

1, then 1 is will goes to 20. So, it’s definitely not commutative, but identity exist what is

identity for permutation, the identity map 1 to 1, 2 to 2, 3 to 3, because we compose

it with any other permutation you get back that same permutation. Inverse? Inverse also

exists. Inverse of a permutation is just an inverse map if we compose an inverse map you

get the identity permutation. More examples. Q star which I use to denote

all nonzero rational numbers under multiplication operation. That is a group. Why is that a

group? Closure holds, associative holds, commutative holds. Identity? 1 is the identity. Inverse?

1 by a. So, if a is a rational number, 1 by a is also a rational number and that is the

inverse. So, this is actually a multiplication operation,

which is different than addition operation for numbers. But if we view it in that abstract

fashion, it is same as the addition operation. This is already a remarkable fact which was

not at all evidence and this becomes evident only when we abstract out the properties of

addition, abstract out properties of multiplication see they are the same.

Multiplication is repeated addition. Multiplication is repeated addition true,

but the fact that properties would remain the same is not at all clear. For example,

if you start with integers. There also multiplication with repeated addition, but over integers

under integers under multiplication do not form a group, because the inverse does not

exist, but over rational they do form a group. So, this is a good time to close, because

we have defined groups, we have given some examples of groups, and already thrown up

certain an unexpected fact. So, what I would like you to do is go back,

think about it. Tomorrow we will meet again at 12, and will continue the discussion.