Venn diagram . In Mathematics and Set Theory it is said that the graphical representation that normally based on circles and rectangles , superimposed or not, illustrates the relations and operations between the sets .
They were devised by the English mathematician and philosopher John Venn around July 1880 with the publication of his work entitled ” On the mechanical and diagrammatic representation of propositions and reasoning ” in the Philosophical Magazine and Journal of Science , which caused quite a stir in the world of formal logic.
Summary
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- 1 Definition
- 2 History
- 3 Examples
- 4 Consequences
- 5 See also
- 6 Sources
Definition
These are the representations according to the Venn diagrams of the different components of set theory and their operations:
| Name | Venn Diagram | Description |
| Universe set | Rectangle labeled with U . Inside all the representations appear. | |
| Set | Circle labeled with the name of the set. | |
| Set | Circle labeled with the name of the set inside. U is excluded when it does not influence the representation and its interpretation. |
|
| Subset | Circle within another circle. | |
| Superset | Circle within another circle. | |
| Complement | TO’. Shading the area outside the set A . | |
| Union | Shading joint areas A and B . | |
| Intersection | ||
| Disjoint sets | Two circles A and B without common areas. | |
| Difference | Shading A of the non – common area with B . AB . | |
| Symmetric difference | Shading non – common areas A and B . |
History
The Venn diagrams are due to John Venn , an English mathematician and philosopher , who around July 1880 published in the Philosophical Magazine and Journal of Science the work entitled ” On the mechanical and diagrammatic representation of propositions and reasonings “, causing a stir in the world of formal logic.
Despite the fact that Leibniz , Boole and Augustus de Morgan had developed previous forms of geometric representation of logical syllogisms, Venn surpassed their systems of representation in clarity and simplicity, eventually becoming the new standard, among other particularities. Venn was the first to formalize their use, offering a mechanism for their generalization.
His book Symbolic Logic , from 1881, although he was not very successful in his efforts to interpret and correct Boole’s works in formal logic ; it became an excellent example platform for the new representation system. Following his next book, “The Principles of Empirical Logic,” from 1889 , Venn diagrams were increasingly used as a representation of logical relationships.
The diagrams of three sets were the most common ones developed by Venn in his initial presentation and although the graphic representation of the universe set was pointed out in his works, its current idea is attributed to Lewis Carroll , mathematician and writer of Alicia in the country of wonders .
Examples
| Conjunct algebra | Venn Diagram |
| A ‘= UA | |
| ( ) – ( ) = | |
| File: Plugin A attached B.gif | |
Consequences
Venn diagrams provide greater readability and understanding of set theory and set algebra. It illustratively contributes to representing relationships that are otherwise more abstract and can thus be shown to lower grade students in general education. They also turned out to be the extension of previous works in the same or similar field of formal logic that in the end transcended due to its simplicity and practical value.
However, deficiencies appear when dealing with algebraic relationships of more than three sets for which solutions such as Edwards extensions have been proposed .
