# Venn diagram

A graphic representation of formulas of mathematical logic, mainly formulas of the propositional calculus. A Venn diagram of $n$ variables $a_1,\dotsc,a_n$ of classical propositional logic is a selection of closed contours $C_1,\dotsc,C_n$ (with homeomorphic circumferences) which subdivides the plane into $2^n$ domains, some of which (e.g. $v_1,\dotsc,v_k$, $0\leq k\leq2^n$) are marked. Each marked domain $v_i$, $0<i\leq k$, is put into correspondence with the formula $B_i=b_1\mathbin{\&}\dotsb\mathbin{\&}b_n$ where $b_j$, $0<j\leq n$, is $a_j$ if $v_i$ lies within the contour $C_j$ and $b_j$ is $\neg a_j$ otherwise. The formula corresponding to the diagram as a whole is $B_1\lor\dotsb\lor B_n$. Thus, the Venn diagram in the figure corresponds to the formula

$$(\neg a_1\mathbin{\&}\neg a_2\mathbin{\&}\neg a_3)\lor(a_1\mathbin{\&}\neg a_2\mathbin{\&}a_3)\lor(\neg a_1\mathbin{\&}a_2\mathbin{\&}\neg a_3).$$

If there are no marked domains ($k=0$), the diagram corresponds to an identically-false formula, e.g. $a_1\mathbin{\&}\neg a_1$. In propositional logic, Venn diagrams are used to solve decision problems, the problem of deducing all possible pairwise non-equivalent logical consequences from given premises, etc. Propositional logic may be constructed as operations over Venn diagrams brought into correspondence with logical operations.

Figure: v096550a

The apparatus of diagrams was proposed by J. Venn [1] to solve problems in the logic of classes. The method was then extended to the classical many-place predicate calculus. Venn diagrams are used in applications of mathematical logic and theory of automata, in particular in solving the problems of neural nets.

#### Comments

The idea of Venn diagrams goes back to L. Euler and they are sometimes also called Euler diagrams.

#### References

[1] | J. Venn, "Symbolic logic" , London (1894) |

[2] | A.S. Kuzichev, "Venn diagrams" , Moscow (1968) (In Russian) |

[a1] | P. Suppes, "Introduction to logic" , v. Nostrand (1957) pp. §9.8 |

[a2] | G. Birkhoff, S. MacLane, "A survey of modern algebra" , Macmillan (1953) pp. 336ff |

[a3] | B. Rosser, "Logic for mathematicians" , McGraw-Hill (1953) pp. 227–228; 237ff |

**How to Cite This Entry:**

Venn diagram.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Venn_diagram&oldid=55919