Математика. Материалы курса

Before you start studying mathematics, it is essential to be acquainted with symbols, with the language of mathematics, because mathematics is a special science, it uses its own symbol language, which has its origins in mathematical logic.

The main concept we use is the concept of a set. This is one of the main undefinable concepts that was first formulated (defined) and became a mathematical object in the 19th century. The German mathematician Georg Cantor, who was born in Russia, is the founder of the set theory.

We use uppercase letters of the Latin alphabet to denote sets, and lowercase letters to denote elements. For example, if a is an element of A, we use the membership sign (Î). If a is not an element of the set A, we say: "a does not belong to A", and the sign is (Ï). To denote an empty set, we also use a special sign, which means that there are no elements in this set (Æ).

Let's start studying symbols. The first thing is the concept of a quantifier. There are two quantifiers: existential ($) and universal ("). Their origin is rather interesting, the symbols are the inverted first letters of the English words Exist and Any. We usually write a variable next to the quantifier. How do we read these characters? There is an x. Note that if you have an existential quantifier next to a variable, then this notation is always followed by an explanation of what element exists. This description applies to this element. If the universal quantifier is next to a variable, we read "for each".

When writing mathematical sentences, we also use logical connectives, symbols of logical operations. The symbol "an inverted cap" (Ù) is a conjunction sign, and we read it as conjunction "and" (aÙb - a and b). If we invert this sign, we get the disjunction sign, and we read it "a or b". What do you need to know about these symbols? They are closely related to other symbols that you encountered in your school math course. The conjunction sign is related to the system sign from the operation of the set intersection. For example, let us consider the double inequation x ≥ 1 and x ≤ 5. This means that the conjunction of two conditions is satisfied, which we can write not using the conjunction sign, but using the sign of the system ({). This means that x belongs to the intersection of solutions of this inequation system. The solution is the segment [1; 5]. As for disjunction, it is associated both with the union sign and with the aggregate record ([). Disjunction of inequations x ≤ 1 or x ≥5 is equivalent to writing the set of these inequations, which in turn means combining two spaces.

The next operation is implication (Þ). If it connects two statements a and b, then we read "if a, then b". Equivalence (ó) is a conjunction of two implications, and we read "a if and only if b".

Finally, a unary operation (we have considered binary operations which link two statements) is negation (ù), it is performed for one statement and in Russian we usually use the particle “ne” which corresponds to the English particle "not".

In our lectures, we also use a special sign, it is similar to the implication, but we read it "by definition means". This helps us to formulate the defined term briefly.

The relationship between the sets (Í). This concept is familiar to you. Note that we usually put an additional horizontal line meaning that sets can be equal. How can we formulate the definition of inclusion (Í) of two sets? AÍBó"x(xÎAÞxÎB): the set B includes the set A. By definition, it means (we need to read quantifiers) that for any x, it is satisfied (then there is the statement, sometimes we enclose it in parentheses, sometimes we do not enclose it), if x belongs to the set A, then x belongs to the set B.

The equivalence relation of sets, in this case, is equivalent to the equality of the conjunction of two inclusions. Sets A and B are called equal if the set A is included in the set B, and the set B is included in the set A.

To understand the material, we should know how to construct the negations of statements, which are built in this logical notation system (see the video).

How do we construct a negation if we want to say that the set A is not included in B? First, what you need to know about quantifiers: when constructing negations, the universal quantifier changes to the existential quantifier and vice versa – the existential quantifier to the universal quantifier. In addition, the utterance form changes its meaning to the opposite; we build a negation. There is a certain rule here. Since we have an implication in the definition of inclusion, we must know how to construct the implication. So, the negation of the implication AÞB (if A, then B) means that the conjunction of statements A and not B is fulfilled. Note how the definition of inclusion is transformed when we move to negation. The set A is not included in the set B means that there is an element x that (we explain which) x belongs to the set A, and x does not belong to the set B.

Let's consider how to construct a negation of the set equivalence. In the definition, we have a conjunction of two conditions, and we need to know how to construct the negation of conjunction and disjunction. We have A and B, we construct a negation, it means either not A, or not B, it does not denote a dividing disjunction, it can be both not A and not B. Thus, the negation of the conjunction A and B is the disjunction of the negatives not A or not B. It is the same with the disjunction negation. What do we do? We break the line, and replace the conjunction sign with the disjunction, and vice versa. In this case, the sets A and B are not equal if A is not included in B, or B is not included in A.

The main numerical sets. They have evolved throughout the history of the development of mathematics. The first set that we use in life is the set of natural numbers: N = {1, 2, 3, ...}. Note that we use the symbol in bold to write in the notation of these numeric sets, because this is an international practice, these are the accepted symbols for denoting sets by the world community of mathematicians. The set of natural numbers is 1 2 3 and so on, the numbers we use to count items. In the set of integers we add zero and the numbers opposite to the natural numbers: Z = {..., -2, -1, 0, 1, 2, ...}.

The set of all rational numbers (Latin for relations, fraction) is the set of all possible fractions m/n, in which m is an integer and n is a natural number: Q = {m/n | mÎZ, nÎN}. The set of irrational numbers is the numbers that are not rational: I. The union of the sets Q and I gives us the entire set of real numbers: R = {-∞; + ∞}, which we identify with the numeric line (we specify the direction, the starting point, and a unit on the line).

In mathematical analysis, we study numerical functions that are related to the concept of a numeric set. What is a numeric set? This is any subset of the set of real numbers, so points on the numeric line represent its elements.

What should we be prepared for? The same entry in mathematics can be read differently. For example, x is contained in R: xÍ R. Is there another way to say it? x is part of the set R, x is a subset of R, but the best variant is to say that x is a numeric set. The entry means exactly this.

In addition, we use not just R, but R². We are not going to go into details why this is so, just let's agree that R² indicates a coordinate plane, the elements of which are pairs of real numbers – these are the points of the coordinate plane. If we have R³, then this is a coordinate three-dimensional space, the elements of this space have three real coordinates.

This concludes the introductory lecture on mathematical symbolism. We wish you a successful study. Besides, we recommend that you take your time, pause the video more often, and delve into the recordings that we offer you.