Mathematics is essentially an abstract science, if we move away from elementary concepts. So, on a couple of apples, you can visually 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? That's the thing, no. The more complex became the concepts with which mathematics operates in its judgments, the more problematic seemed 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 derived, examples and possibilities of which we will consider below.
A bit of history
On April 17, 1707, the world gave science Leonhard Euler, a remarkable scientist whose contribution to mathematics, physics, shipbuilding and even music theory cannot be overestimated.
His works are recognized and in demand all over the world to this day, despite the fact that science does not stand still. Of particular interest is the fact that Mr. Euler took a direct part in the formation of the Russian school of higher mathematics, especially since, by the will of fate, he returned to our state twice. The scientist had a unique ability to build algorithms that were transparent in their logic, cutting off everything superfluous and moving from the general to the particular in the shortest possible time. We will not list all his merits, since it will take a considerable amount of time, and we will turn directly to the topic of the article. It was he who suggested using a graphic representation of operations on sets. Euler circles are able to visualize the solution of any, even the most complex problem.
What is the point?
In practice, Euler circles, the scheme of which is shown below, can be used not only in mathematics, since the concept of "set" is inherent not only in this discipline. So, they are successfully applied in management.
The diagram above shows the relations of sets A (irrational numbers), B (rational numbers) and C (natural numbers). The circles show that set C is included in set B, while set A does not intersect with them in any way. The example is the simplest, but it clearly explains the specifics of "relationships of sets", which are too abstract for real comparison, if only because of their infinity.
Algebra of logic
This areamathematical 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 true, while the last is false. 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, examples with their use are too convenient and visual to be ignored.
A bit of theory:
- Let 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 sets A and B consists of elements that belong simultaneously to both set A and set B.
- The union of sets A and B consists of elements that belong to set A or set B.
- The negation of set A is a set that consists of elements that do not belong to set A.
All this is depicted again by Euler circles in logic, since with their help each task, regardless of the degree of complexity, becomes obvious and visual.
Axioms of the algebra of logic
Assume that 1 and 0 exist and are defined in set A, then:
- negation of the negation of set A is set A;
- union of set A with not_A is 1;
- union of set A with 1 is 1;
- union of set A with itself is set A;
- union of set Awith 0 there is a set A;
- intersection of set A with not_A is 0;
- the intersection of set A with itself is set A;
- intersection of set A with 0 is 0;
- the intersection of set A with 1 is set A.
Basic properties of the algebra of logic
Let sets A and B exist and are not empty, then:
- for the intersection and union of sets A and B, the commutative law applies;
- the combination law applies to the intersection and union of sets A and B;
- distributive law applies to the intersection and union of sets A and B;
- the negation of the intersection of sets A and B is the intersection of the negations of sets A and B;
- the negation of the union of sets A and B is the union of the negations of sets A and B.
The following shows Euler circles, examples of intersection and union of sets A, B and C.
Prospects
Leonhard Euler's works are reasonably considered the basis of modern mathematics, but now they are successfully used in areas of human activity that have appeared relatively recently, take at least corporate governance: Euler's circles, examples and graphs describe the mechanisms of development models, be it Russian or English-American version.
