Euler circles: examples and possibilities

Mathematics is inherently abstract science, if we move away from elementary concepts. So, on a pair of three apples you can clearly visualize the basic operations that underlie mathematics, but as the plane of activity expands, these objects become insufficient. Has anyone tried to depict operations on infinite sets on apples? That's just the point, that no. The more complicated the concepts that mathematics operates in its judgments, the more problematic it seemed to be their visual expression, which would be designed to facilitate understanding. However, for the happiness of both modern students and science as a whole, Euler circles were derived, examples and possibilities of which we will consider below.

A bit of history

On April 17, 1707, the world presented science to Leonhard Euler, a remarkable scientist whose contributions to mathematics, physics, shipbuilding and even music theory were not overestimated. His works are recognized and in demand to this day in the whole world, despite the fact that science does not stand still. Particularly interesting is the fact that Mr. Euler took a direct part in the formation of the Russian school of higher mathematics, especially as he twice returned to our state by the will of fate. The scientist possessed a unique ability to build algorithms transparent in his logic, cutting off all unnecessary and moving from general to particular in the shortest possible time. We will not list all his merits, since it will take a considerable amount of time, and turn directly to the topic of the article. It was he who suggested using a graphic representation of operations on sets. Circles Euler the decision of any, even the most difficult task, can be depicted visually.

What is the essence?

In practice, Euler circles, the scheme of which is depicted below, can be applied not only in mathematics, since the concepts of "set" are inherent not only in this discipline. So, they are successfully applied in management.

The diagram above shows the relations of the sets A (irrational numbers), B (rational numbers) and C (natural numbers). The circles show that the set C is included in the set B, while the set A does not intersect with them in any way. The simplest example, but clearly explains the specifics of the "interrelationships of sets", which are too abstract for real comparison, if only because of their infinity.

Algebra of logic

This area of mathematical logic operates with statements that can be both true and false. For example, from the elementary: the number 625 is divided by 25, the number 625 is divided by 5, the number 625 is simple. The first and second statements are truth, while the latter is a lie. Of course, in practice everything is more complicated, but the essence is shown clearly. And, of course, Euler circles are again involved in the solution, the examples with their use are too convenient and obvious to ignore.

A bit of theory:

• Let the sets A and B exist and not be empty, then for them the following operations of intersection, union and negation are defined.
• The intersection of the sets A and B consists of elements that belong simultaneously to both the set A and the set B.
• The union of the sets A and B consists of elements that belong to the set A or to the set B.
• Denial of a set A is a set that consists of elements that do not belong to the set A.

All this again depicts Euler's circles in logic, since with their help each problem, regardless of the degree of complexity, becomes obvious and obvious.

Axioms of the algebra of logic

Suppose that 1 and 0 exist and are defined in the set A, then:

• The negation of the negation of A is the set A;
• The union of the set A with non-A is 1;
• The union of the set A with 1 is 1;
• The union of A with itself is the set A;
• The union of the set A with 0 is the set A;
• The intersection of A with non-A is 0;
• The intersection of A with itself is the set A;
• The intersection of A with 0 is 0;
• The intersection of the set A with 1 is the set A.

Basic Properties of Algebra of Logic

Suppose that the sets A and B exist and are not empty, then:

• For the intersection and union of the sets A and B, a traveling law operates;
• For the intersection and unification of the sets A and B, a combining law operates;
• For the intersection and unification of the sets A and B, the distribution law applies;
• The negation of the intersection of the sets A and B is the intersection of the negations of the sets A and B;
• The negation of the union of the sets A and B is the union of the negations of the sets A and B.

Below we show Euler's circles, examples of intersection and union of sets A, B and C.

Prospects

Leonard Euler's works are reasonably considered to be the basis of modern mathematics, but now they are successfully applied in the fields of human activity that have appeared relatively recently, to take at least corporate governance: Euler circles, examples and graphs describe the mechanisms of development models, whether Russian or Anglo-American version .