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.

🛠️ This page contains images that should be replaced by better images in the SVG file format. 🛠️

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