Leonhard Euler (1707-1783) - known Swiss and Russian mathematician, a member of the St. Petersburg Academy of Sciences, most of his life in Russia.The most well-known in mathematical analysis, statistics, computer science and logic, is considered a circle Euler (Euler-Venn diagram) is used to denote the scope of the concepts and sets of elements.
John Venn (1834-1923) - English philosopher and logician, co-author of the Euler-Venn diagrams.
compatible and incompatible concepts
The term refers to a form of logic thinking, reflecting the essential features of a class of similar items.They are identified by one or a group of words, "world map", "The Dominant kvintseptakkord", "Monday" and others.
In the case where the volume elements of the concept of wholly or partially owned by another volume, suggests compatible concepts.If none of the volume element defined concept does not belong to the scope of the other, we have a place with incompatible concepts.
In turn, each of the types of concepts has its own set of possible relationships.For compatibility of concepts is the following:
- identity (equivalence) volumes;
- intersection (overlap) volumes;
- submission (subordination).
to incompatible:
- subordination (coordination);
- contrast (contraries);
- contradiction (kontradiktornost).
Schematically, the relationship between the concepts of logic can be designated using the Euler-Venn circles.
Relationship equivalency
In this case, the concept of mean the same thing.Accordingly, the scope of these concepts are the same.For example:
A - Sigmund Freud;
B - founder of psychoanalysis.
Either:
A - square;
B - equilateral rectangle;
C - equiangular rhombus.
used to indicate exactly match the circles of Euler.
intersection (overlap)
This category includes the concept of sharing common elements found in regard to the crossing.That is the amount of one of the concepts is partially included in the scope of another:
A - teacher;
B - music fan.
As can be seen from this example, the volume of concepts overlap: a certain group of teachers may be music lovers, and vice versa - among music fans may be representatives of the teaching profession.Similarly, the ratio will be the case when, as a concept and a speaker, for example, "citizen," and for B - "autodriver".
Submission (subordination)
schematically indicated as different scale Euler circles.The relationship between the concepts in this case characterized by the fact that the concept of subordinate (lower volume) is completely a part of subordinates (high volume).This does not exhaust the concept of slave fully comply.
example:
A - tree;
B - pine.
The concept will be subordinate to the concept of A. Since pine belongs to the trees, the concept of A becomes in this example to obey, "absorb" the amount of concepts B.
This subordination (coordination)
relations are characterized by two or more conceptswhich exclude each other, but at the same time belonging to a certain circle of common descent.For example:
A - clarinet;In
- guitar;
C - violin;
D - a musical instrument.
Concepts A, B, C are not overlapping with respect to each other, however, they all belong to the category of musical instruments (the term D).
opposite (contraries)
opposite relationship between the concepts of mean relatedness of these concepts to the same genus.Thus one of the concepts has certain properties (attributes), while their other denies replacing opposite in character.Thus, we are dealing with antonyms.For example:
A - dwarf;
B - giant.
Euler diagram for opposite relationship between concepts is divided into three segments, the first of which corresponds to the concept A, the second - in the concept, and the third - the rest of the possible concepts.
contradiction (kontradiktornost)
In this case, both concepts are views of the same kind.As in the previous example, one of the concepts indicates certain properties (attributes), while their other rejects.However, unlike the relationship opposite second opposite concept does not replace denied properties other alternative.For example:
A - a difficult task;
in - simple task (non-A).
Expressing the scope of the concepts of this kind, Euler diagram is divided into two parts - the third, an intermediary in this case does not exist.Thus, the concepts are also the antonyms.At the same time one of them (A) becomes positive (approving any indication) and the second (B or A) - negative (denying the appropriate sign), "White Paper" - "not the white paper", "national history"- "foreign history" and so on. d.
Thus, the ratio of the concept in relation to each other is a key characteristic that defines the range of Euler.
Relations between sets
should also distinguish between elements and sets, the volume of which reflect Euler circles.The concept is borrowed from the set of the mathematical sciences and has a fairly broad meaning.Examples of logic and math displays it as a kind of collection of objects.Themselves objects are elements of the set."A lot has much as a conceivable" (Georg Cantor, the founder of the theory of sets).
designation sets is carried out in capital letters: A, B, C, D ... and so on. G., Elements of sets - in lower case: a, b, c, d ..., and others. Examples of sets may be students who are inone audience, books, standing on a certain shelf (or, for example, all the books in a certain library), the pages in the diary, berries in a forest glade, and so on. d.
In turn, if a certain set does not contain anyelement, it is called empty and indicate the sign Ø.For example, the set of points of intersection of parallel lines, the set of solutions of the equation x2 = -5.
Problem Solving
to solve a large number of tasks widely used Euler circles.Examples of logic demonstrate the connection of logic operations with the theory of sets.It uses the concepts of truth table.For example, a circle, indicate the name of A is the area of truth.Thus, the area outside the circle will be a lie.To determine the area of the chart for a logic operation should be hatched area defining Euler diagram in which its values for the elements of A and B are true.
Using Euler diagram found wide application in various industries.For example, in a situation with a professional choice.If the subject is concerned about choosing a future profession, he can be guided by the following criteria:
W - what I like to do?
D - that I get?
P - than I can make good money?
portray it as a scheme: Euler diagram (examples in logic - the ratio of the intersection):
result will be those professions that will be at the intersection of the three circles.
special place following Euler-Venn take mathematics (set theory) in the calculation of combinations and properties.Euler diagram of the set of elements enclosed in the image rectangle indicating the universal set (U).Instead of circles can also be used other closed figures, but the essence remains the same.The figures intersect, according to the terms of the problem (in the most general case).Also, these figures should be marked accordingly.As elements of the sets under consideration may serve points located within the different segments of the chart.On this basis, you can hatch a specific area, denoting thereby newly created sets.
With data sets permitted to perform basic mathematical operations: addition (sum of sets of elements), subtraction (difference), multiplication (product).In addition, thanks to the Euler-Venn diagrams can perform operations on the number of sets of comparison of their constituent elements, not counting them.
