Venn diagram

Pie charts are used for graphical illustration of set theory. There are different types of quantity charts, in particular Euler diagrams ( after Leonhard Euler ) and Venn diagrams (after John Venn ).

Pie charts can illustrate quantitative relationships, but are not suitable as a mathematical proof in general. As evidence, are suitable only those pie charts that represent all the possible relations of the quantities represented; Such diagrams are called Venn diagrams. The disadvantage of Venn diagrams is that they are rapidly confusing with more than three quantities involved because they must represent at n objects 2n possibilities. Venn himself was under the use of ellipses represent up to four, and finally even five quantities involved.

Examples

Euler diagrams

Euler diagrams are used primarily to make set-theoretic facts, for example, the subset property, vividly. The following illustrations are common:

; is a member of.

; is not an element of.

Respectively; is a subset of or is superset of.

Venn diagrams

Venn diagrams show all the relations between the observed quantities dar. Therefore, we can use them to read relationships and infer from the presence of individual relations on the existence of other relations.

( Average); A cut B, that is, all elements which are contained both in A and in B.

( Union ); A combined with B, that is, all elements which are included in A or B or both.

(Difference amount ); A without B, that is, all elements which are contained in A, but not in B. With the negation is written:

( Symmetric difference ); not A or B, that is, all elements which are included in A or B but not both. notation:

( Complement of A);   contains all the elements of the universe U are not in A

Venn diagrams are mainly known in the representation of three sets with circles. However, Venn had the ambition to find " symmetrical figures in elegant high " that represent a larger number of volumes, and showed a diagram for four sets in an elliptical shape. He then gave a construction method by which one can represent Venndiagramme for an " arbitrary " number of sets, each closed curve is intertwined with the other, starting from the diagram with three circles. This is a "tube" drawn over the most recent set representation. This means that all other quantities are cut.

Johnston diagrams

Johnston diagrams are a bivalent propositional logic interpretation of quantity charts, specially Venn diagrams. In Johnston Diagram, a circle is interpreted ( a lot ) P as the set of circumstances under which a proposition P is true. The area outside of the circle ( the complement of the set ) P is interpreted as the set of circumstances under which the statement is false. To say that a statement is true, you paint the entire area outside their circle of black; it thus indicates that the facts under which the statement is not true, may not apply. Conversely, to say that a statement is false, you paint the area within their circle of black; they say that the facts under which the statement is true, may not apply. Combining two statements P, Q by a conjunction, that is, one wants to express that both statements are true, you paint the entire area that is outside the intersection of the circles P, Q, black to; they say that none of the circumstances under which not both P and Q are true, may be present.

Johnston diagrams are thus an image of the classical propositional logic to the basic set theory, where negation as complementation, the conjunction as intersection and disjunction can be represented as a union. The truth values ​​true and false are mapped to the Allmenge or to the empty set.

History

Leibniz was already using the 1690 amount charts showing the syllogistic. Christian way, Rector of the Gymnasium in Zittau, used around 1700 pie charts to represent logical operations. JC Lange published in 1712 the book Nucleus Logicae Weisianae, is treated in the Sage logic. Leonhard Euler, Swiss mathematician in the 18th century, introduced the Euler diagram, which he first used in a letter dated February 24, 1761.

John Venn, British mathematician in the 19th century, led 1881 the Venn diagram. 1964 works of Peirce are first recognized academic who had this written in the last quarter of the 19th century and describe the existential graphs.

The following graphics show how Venn diagrams are used to illustrate syllogisms since the 17th century. The validity of an inference can be verified using this method. ( Instance, could see that the Darapti mode is valid only on the assumption of a non-empty middle term. )

In black areas no element exists ( universal statement ). In red areas there is at least one element x ( existence theorem ).

Such Venn diagrams are easy to Euler diagrams forming, as the following chart shows. Venn diagrams have the advantage that you can forget about any overlap, so they are also suitable for evidence. Contrast, can be more intuitive grasp of the quantities inside one another or overlap with Euler diagrams.