1.1. Definitions and examples of sets

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From the course by National Research University Higher School of Economics

Mathematics for economists

10 ratings

National Research University Higher School of Economics

Mathematics for economists

10 ratings

This course is an important part of the undergraduate stage in education for future economists. It's also useful for graduate students who would like to gain knowledge and skills in an important part of math. It gives students skills for implementation of the mathematical knowledge and expertise to the problems of economics. Its prerequisites are both the knowledge of the single variable calculus and the foundations of linear algebra including operations on matrices and the general theory of systems of simultaneous equations. Some knowledge of vector spaces would be beneficial for a student. The course covers several variable calculus, both constrained and unconstrained optimization. The course is aimed at teaching students to master comparative statics problems, optimization problems using the acquired mathematical tools. Home assignments will be provided on a weekly basis. The objective of the course is to acquire the students’ knowledge in the field of mathematics and to make them ready to analyze simulated as well as real economic situations. Students learn how to use and apply mathematics by working with concrete examples and exercises. Moreover this course is aimed at showing what constitutes a solid proof. The ability to present proofs can be trained and improved and in that respect the course is helpful. It will be shown that math is not reduced just to “cookbook recipes”. On the contrary the deep knowledge of math concepts helps to understand real life situations.

From the lesson

The basics of the set theory. Functions in Rn

Week 1 of the Course is devoted to the main concepts of the set theory, operation on sets and functions in Rn. Of special attention will be level curves. Also in this week introduced definitions of sequences, bounded and compact sets, domain and limit of the function. Also from this week students will grasp the concept of continuous function.

1.1. Definitions and examples of sets14:35

1.2. Operations on sets9:31

1.3. Open balls in Rn9:10

1.4. Sequences in Rn. Closed sets.10:33

1.5. Bounded and compact sets10:48

1.6. Functions and level curves in Rn16:47

1.7. Domain and limit of a function. Continuous functions.13:51

1.8. Continuity of a function. Weierstrass theorem.13:04

1.9. Composite function.12:07

1.10. Continuity of a composite function3:30

Meet the Instructors

Kirill Bukin
Associate Professor, Candidate of sciences (phys.-math.)
Faculty of Economic Sciences, Department of Theoretical Economics

We start this class with a set theory or rather we would like

to remind you the basic concepts, terminology, and notation.

Usually, in elementary math,

we deal with sets,

which are sets of real numbers.

So, when we write a capital letter representing a set,

it can be a so-called open interval,

where a, b are real numbers.

We'll also use a notation "Belongs to," so

this open interval is a subset or the set of real numbers.

So, this is a notation describing the set of real numbers.

Along with this open interval,

we can also use

closed intervals described like that,

and I would like to remind you that what other iterations on sets.

Maybe I'll provide a numerical example.

So, let's suppose we have an open interval,

which is I'm using braces to describe

this set consists of real numbers and

the element from this set is x.

This symbol denotes "Belongs to" the set of real numbers

and the vertical bar has the meaning of "Such that."

So, open interval, which includes all real numbers from

1 to 3 can be written using double inequality.

Another open interval from 2 to

4 can be written analogously.

Now,

I would like

to unite these two sets.

So, I will be using a special symbol.

So, I start with the first set from 1 to 3.

Now, I use the symbol or the union,

which looks like a capital letter "U," followed by second set from 2 to 4.

Then, what do I get.

It's quite useful to use geometrical representation, number line.

So, if we draw the number line where somewhere we have the origin.

So, this is the point where x equals 0.

So here, we can draw or plot the numbers,

1, 2, 3, 4.

So, the first set from 1 to 3 is the set ranging from 1.

This is the left endpoint.

By the way, we have strict inequalities here, which tells us.

This information tells us that neither 1 or 3 are included,

they don't belong to and the second interval ranges from 2 to 4.

Now, we unite them, so what do we get?

A union means that this is a new set where elements

belong to either the first set or the second set or maybe to both.

So, that makes the answer,

this is a set ranging from 1 to 4.

If we turn upside

down this "U" letter,

we'll get another symbol,

a symbol of intersection.

So here, we will see what do we get.

This is my symbol I'm intersecting

the same 2 open intervals and

this intersection means that it's made of elements where such x,

which belongs to this intersection belongs to the first and the second.

So usually, how do we say a union?

X belongs to the Union.

If x belongs either to the first or the second or to both,

here we'd say it belongs to the first and to the

second and here I will shade the intersection.

So the answer is from two to three.

Along with the operations like union and intersection,

we have the third operation which is called finding complements of sets.

Let's suppose D is a set of real numbers.

Now, we'll call complement of D and we use this symbol,

this is a horizontal horizontal bar above the letter, that's a definition.

This set is made of all real numbers such that, remember vertical bar,

they do not belong to D. So, for example,

if I choose this interval from one to three,

I would like to find its complement.

How will it look like?

I would like to use the same symbols, round brackets,

so it is made of all real numbers which are not included.

So, I start with a negative infinity.

Here, use the square bracket,

a union symbol, a new square bracket, positive infinity.

So, such intervals are called,

we can use half-open or half-closed.

So that means that number one is included in this set,

the same true for number three.

Now, we proceed with

the sets from the n-dimensional space.

So, we need to explore what is an n-dimensional space.

Our interests will be in R_n,

and the letter R

which we used earlier is

a particular example of an n-dimensional space when n equals one.

In courses like linear algebra,

there is such a notion as a vector space and this is a case of a vector space.

So, the elements of such vectors,

usually in texts, bold letters are used to describe vectors

but we'll be using the same letter X.

So, X from belongs to

n-dimensional space is a vector represented by a column.

This column consists of n coordinates starting from the first,

and clearly since I don't know n,

I'm using dots, and the final coordinate is X_nth.

So, this is a vector from R_n,

we call such a space.

What are operations which we can perform on vectors in this space?

According to the definition of a vector space,

there are two operations.

The first is a sum of two vectors so we need to define how to vectors are added up.

The second operation is a scalar multiplication;

when any vector can be multiplied by a scalar which is actually a real number.

So, let's suppose we have two vectors,

the first being x, already written here,

the second is y.

Also, this vector is a column which includes n coordinates,

three dots, and probably many, many more.

So, how two vectors,

x and y, are added up?

They're added up coordinate by coordinate.

So, when we add two vectors,

that means that we form

the resulting vector where the first coordinate of

x is added with the first coordinate of y,

the same true for all other coordinates.

Writing is always three dots here and that's the nth coordinate.

Now, multiplication by scalar.

What is a scalar? Scalar is a real number.

How to multiply a vector,

x, by the scalar?

Again, coordinate by coordinate.

So, that's the definition

and the final.

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