Mathematics inherently abstract science, if you move away from the basic concepts.For example, in a pair of triple apples you can graphically depict the basic operations that are the basis of mathematics, but as soon as the plane of activity expands, these objects are becoming scarce.Somebody tried to portray on apples operations on infinite sets?The fact of the matter is that there is.The more complex the idea that mathematics operates in the judgments, the more problematic it seemed their visual expression, which would be designed to facilitate understanding.However, today's happiness as students, and science in general have been withdrawn following Euler, examples and opportunities which we discuss below.
little history
April 17, 1707 gave the world the science of Leonhard Euler - outstanding scientist whose contributions to mathematics, physics, shipbuilding and even music theory not be overestimated.His works are recognized and in demand to this day around the world, despite the fact that science does not stand still.Particularly amusing is the fact that Mr. Euler was directly involved in the development of the Russian school of higher mathematics, the more so as the fates decree, he twice returned to our country.The scientist had a unique ability to build transparent in its logic algorithms, cutting off all unnecessary and quickly moving from the general to the particular.We will not list all his achievements, because it will take a considerable amount of time and turn directly to the subject of the article.It was he who suggested the use of a graphic representation of operations on sets.Euler diagram solution to any, even the most difficult tasks drawn up, can depict visually.
What is the essence?
In practice, following Euler diagram is shown below can be used not only in mathematics, since the concept of "plurality" are not unique to the discipline.So, they have been successfully applied in management.
diagram above shows the relationship set a (irrational number), B (rational numbers) and C (integers).Circles indicate that the set is included in the set B, whereas A lot of them does not intersect.An example of a simple but clearly explains the specifics of "relationship sets" that are too abstract for a real comparison, if only because of their infinity.
algebra of logic
This area of ​​mathematical logic operates statements, which can be both true and false character.For example, from the elementary: The number 625 is divisible by 25, the number 625 is divisible by 5, the number 625 is simple.The first and second approval - the truth, while the latter - a lie.Of course, in practice, more complex, but the essence is shown clearly.And, of course, once again participate in the decision Euler diagram, examples of their use is too convenient and intuitive to ignore.
little theory:
- Let the sets A and B, and there are not empty, then for them, the following operations intersection, union and negation.
- intersection of sets A and B consists of the elements that belong to both a set A and set B.
- Union of sets A and B consists of the elements that belong to the set A or set B.
- Denial of A - is a set ofthat consists of elements that do not belong to the set A.
All this is portrayed again Euler diagram in logic, since they help each task, regardless of the degree of complexity becomes apparent and visible.
Axioms of algebra of logic
Suppose that 1 and 0, there are determined in a variety of A, then:
- negation of the negation of A is the set of A;
- association of A with ne_A have 1;
- Association of A 1 has one;
- association of A with itself is the set of A;
- Association of A 0 is the set of A;
- the intersection of A with ne_A have 0;
- the intersection of A with itself is the set of A;
- the intersection of A with 0 is 0;
- intersection of A 1 is the set A.
basic properties of the algebra of logic
Let the sets A and B, and there are not empty, then:
- for intersection and union of sets A and B acts commutative law;
- for intersection and union of sets A and B acts associative law;
- for intersection and union of sets A and B acts distributive law;
- denial of the intersection of sets A and B is the intersection of negations of A and B;
- denial of the union of sets A and B is the union of negatives sets A and B.
Euler diagram below shows examples of the intersection and union of sets A, B and C.
Prospects
works of Leonhard Euler considered reasonable basis of modern mathematicsbut now they are successfully used in the fields of human activity that are relatively new, at least to take corporate governance: Euler diagram, examples and charts describe the mechanisms of development models, whether Russian or Anglo-American version.
