__<<<__|

__one__|

__2__|

**3**|

__four__|

__five__|

__6__|

__7__| ... |

__17__|

__>>>__

What does that mean is a subset of the set ? This means that all elements of the set belong and set . If you imagine sets in the form of boxes, then - this is a big box, and many - a smaller box, in which some of the elements lying in the box lie . Designation: .

For example, the set of all even numbers is a subset of the set of all integers, and the set - a subset of the set .

Consider two sets:

all flying crocodiles and all participants of the Olympiad .

Is one of them a subset of the other?

How to prove that ? You can check that any item sets lies in . And you can apply the method by contradiction __ * 2 __ : if a not a subset then there is an element such that and if so no then .

__* 2__Opposite, ugly, ugly ...

But is it possible to find a flying crocodile, not participating in the Olympiad? But where can you even find a flying crocodile ... Therefore

all flying crocodiles all participants of the Olympiad __ * 3 __ .

__* 3__What happens: all flying crocodiles participate in the Olympiad?

The set of flying crocodiles is an *empty* set: there are no elements in it. This set is so important that for it even came up with a special symbol: __ *four __ . There is only one symbol for the empty set, because the empty set is unique. Indeed, suppose that there are two different empty sets. But what means that the sets are different? This means that in one of them there is an element that does not belong to the other. But in empty sets there are no elements at all!

__*four__And programmers have stolen this symbol and used to denote zero.

So, we have proved that the empty set is unique and is a subset of any other set.

__<<<__|

__one__|

__2__|

**3**|

__four__|

__five__|

__6__|

__7__| ... |

__17__|

__>>>__

## The similar

Mikhail Raskin

Modern mathematics uses set theory as its foundation. Traditionally, when analyzing set-theoretic subtleties, Zermelo-Fraenkel axiomatics with the axiom of choice, denoted ZFC, is used. The axiom of choice is based on the proof of the existence of a basis in any vector space and the existence of an immeasurable set in mathematical analysis. Unfortunately, the theory of sets must work with sets that are not described in sufficient detail and concretely so that we can imagine them. The course will consider one example of what this leads to. It turns out that at the cost of weakening the axiom of choice, one can obtain set theory in which any bounded function on an interval is Lebesgue integrable. The fact that the axiom of choice is used, in a sense, has happened historically. The course is based on R.M. Solovay on the construction of set theory, in which all the sets of real numbers are measurable.

Mikhail Raskin

In set theory, there are several well-known questions about whether another axiom follows from some axioms (or a hypothesis; an axiom is just a hypothesis that is used by the overwhelming majority). As in other areas of mathematics, unprovability can be demonstrated using a model in which the assumptions are correct, but the hypothesis is not true. To build one of the most famous such examples, the model of set theory, in which there is an intermediate power between the powers of the natural series and the real line, Cohen developed a forcing method.

Viktor Viktorov

Basic concepts, operations on sets, identities, properties of a complement, De Morgan's rule, properties of a symmetric difference; mapping (function), factorization, equivalence relation, barber paradox; ordered sets, minimal, smallest, maximal, and largest elements in an ordered set, majorant and minorant; axiom of choice, a well-ordered set.

Proskuryakov I.V.

The purpose of this book is the strict definition of numbers, polynomials and algebraic fractions and the justification of their properties already known from school, and not familiarizing the reader with new properties. Therefore, the reader will not find here new facts for him (except, perhaps, some properties, real and complex numbers), but he will find out how things are proved that are well known to him, starting with “two and two - four” and ending with the rules of actions with polynomials and algebraic fractions. But the reader will get acquainted with a number of common concepts that play a major role in algebra.Gick E. Ya.

Peter Atkins

Smallian raymond

Vladimir Arnold

Smallian raymond

__Further >>>__Is one of them a subset of the other?

But is it possible to find a flying crocodile, not participating in the Olympiad?

What happens: all flying crocodiles participate in the Olympiad?

But what means that the sets are different?