# Venn diagram

A graphic representation of formulas of mathematical logic, mainly formulas of the propositional calculus. A Venn diagram of $n$ variables $a_1,\ldots,a_n$ of classical propositional logic is a selection of closed contours $C_1,\ldots,C_n$ (with homeomorphic circumferences) which subdivides the plane into $2^n$ domains, some of which (e.g. $v_1,\ldots,v_k$, $0\leq k\leq2^n$) are marked. Each marked domain $v_i$, $0<i\leq k$, is put into correspondence with the formula $B_i=b_1\&\ldots\&b_n$ 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 $B_1\lor\ldots\lor B_n$. Thus, the Venn diagram in the figure corresponds to the formula

$$(\neg a_1\&\neg a_2\&\neg a_3)\lor(a_1\&\neg a_2\&a_3)\lor(\neg a_1\&a_2\&\neg a_3).$$

If there are no marked domains ($k=0$), the diagram corresponds to an identically-false formula, e.g. $a_1\&\neg a_1$. 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.

#### References

 [1] J. Venn, "Symbolic logic" , London (1894) [2] A.S. Kuzichev, "Venn diagrams" , Moscow (1968) (In Russian)