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.
