Mathematics, in its essence, is an abstract science, if we move away from elementary concepts. So, on a couple of apples you can clearly depict the basic operations that underlie mathematics, but as soon as the plane of activity expands, these objects become insufficient. Has anyone tried to depict operations on infinite sets on apples? The fact of the matter is that no. The more complex the concepts that mathematics operates in their judgments become, the more problematic is their visual expression, which would be designed to facilitate understanding. However, for the happiness of both modern students and science in general, Euler circles were deduced, examples and capabilities of which we will consider below.
A bit of history
On April 17, 1707, the world presented science to Leonard Euler, a remarkable scientist whose contribution to mathematics, physics, shipbuilding, and even music theory cannot be overestimated.
His works are recognized and demanded to this day throughout the 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, all the more so because, by the will of fate, he twice returned to our state. The scientist had a unique ability to build algorithms transparent in his logic, cutting off all unnecessary and in the shortest possible time moving from general to particular. We will not list all his merits, since it will take a considerable amount of time, and we turn directly to the topic of the article. It was he who suggested using a graphical representation of operations on sets. Euler’s circles can solve any, even the most difficult task, visually.
What is the point?
In practice, Euler circles, the scheme of which is depicted below, can be used not only in mathematics, since the concepts of "set" are inherent not only in this discipline. So, they are successfully used 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 "relationship 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 divisible by 25, the number 625 is divisible by 5, the number 625 is prime. The first and second statements are truth, while the last is a lie. Of course, in practice, everything is more complicated, but the essence is shown clearly. And, of course, Euler circles again participate in the solution, examples with their use are too convenient and obvious to ignore.
A bit of theory:
- Let the sets A and B exist and are not empty, then the following operations of intersection, union, and negation are defined for them.
- 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 the set B.
- The negation of the set A is a set that consists of elements that do not belong to the set A.
All this is again depicted by Euler circles in logic, since with their help each task, regardless of the degree of complexity, becomes obvious and obvious.
Axioms of algebra of logic
Suppose that 1 and 0 exist and are defined in the set A, then:
- the negation of the negation of the set 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 the set A with itself is the set A;
- the union of the set A with 0 is the set A;
- the intersection of the set A with non_A is 0;
- the intersection of the set A with itself is the set A;
- the intersection of the set A with 0 is 0;
- the intersection of the set A with 1 is the set A.
Basic properties of the algebra of logic
Let the sets A and B exist and are not empty, then:
- for the intersection and union of sets A and B, a transitive law applies;
- for the intersection and union of sets A and B, a combination law applies;
- for the intersection and union of sets A and B, a distribution law applies;
- the negation of the intersection of the sets A and B is the intersection of the negatives 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 are Euler circles, examples of intersection and union of sets A, B and C.
Prospects
Leonard Euler’s works are justifiably considered the basis of modern mathematics, but now they are successfully used in areas of human activity, which appeared relatively recently, to take at least corporate governance: Euler’s circles, examples and graphs describe the mechanisms of development models, whether it’s the Russian or Anglo-American version .