Set theory was born out of a need to put mathematics on a firm, logical foundation. The language of set theory allowed mathematicians to give precise definitions, replacing the unsatisfactory and fuzzy language of the seventeenth century with the much more exact language of sets. To illustrate the concepts of set theory, different types of logical diagrams emerged, the most famous being the well-known Venn diagrams.
It was John Venn (1834-1923) who invented the famous diagram that bears his name. He was born in England and attended Gonville and Caius College Cambridge. As an undergraduate he distinguished himself in mathematics. He graduated in 1857 and, following family tradition, was ordained as a priest two years later. In 1862 he returned to Cambridge as Lecturer of Moral Sciences where he taught philosophy and logic to undergraduates. He eventually resigned from the priesthood.
John Venn wrote several books, including Symbolic Logic in 1881. He was very interested in history. He collaborated with his son in writing an extensive history of Cambridge. He also researched his own family history to write The Annals of a Clerical Family. He died at the age of 88. According to the obituary in the Times, "he was a kindly, friendly man, never ruffled, and he retained his bodily and mental activity to the last".
A precursor to Venn diagrams, called an Euler diagram, had been used by Leonhard Euler (1707-1783) to symbolically represent a logical proposition. For example, the following diagram appears in Euler's Opera Omnia:
Euler uses this diagram to illustrate the syllogism(1)
Translating Euler's statements into the language of sets, we have(2)
Venn was certainly familiar with Euler diagrams but found them inadequate to illustrate general relationships among sets or propositions. As Venn put it:
The weak point about these [Eulerian circles] consists in the fact that they only illustrate in strictness the actual relations of classes to one another, rather than the imperfect knowledge of these relations which we may possess, or wish to convey, by means of the proposition. (From Symbolic Logic, Chapter 2)
Thus, whereas Euler's diagram for three sets does not contain a region for the intersection of $B$ and $C$ (since this region is empty according to his premise that "No $B$ is $C$"), a Venn diagram for three sets $A$, $B$, and $C$, must contain regions corresponding to all possible intersections of the three sets and their complements. The familiar Venn diagram for three sets is shown below:
The eight regions, numbered $0,1,2,\dots 7$ correspond to the possible intersections $XYZ$ where $X$ is either $A$ or its complement $A'$, $Y$ is either $B$ or $B'$, and $Z$ is either $C$ or $C'$. To use this diagram to illustrate the statement "All $A$ is $B$", we need to convey that regions 1 and 5 are empty. Venn did this by shading regions 1 and 5. Similarly, to illustrate that "No $B$ is $C$ and no $C$ is $B$", we would shade regions 6 and 7, since these must be empty.
Now we see that regions 5 and 7 are both empty, so we can draw the conclusion that no $C$ is $A$. Hence Venn used this diagram to prove the validity of the syllogism (1).
Venn used his four-set diagram shown below to illustrate how to solve several logic problems.
|Figure 1: Venn's Four-Set Diagram|
What about Venn diagrams for more than four sets? Venn came up with the following diagram for five sets. The fifth set, which is shaded, is in the shape of a doughnut.
|Figure 2: Venn's Five-Set Diagram|
1. Problem (from John Venn's Symbolic Logic): You are given the following four statements:
|Every $Y$ is either $X$ and not $Z$, or $Z$ and not $X$.|
|Every $WY$ is either both $X$ and $Z$ or neither of the two.|
|All $XY$ is either $W$ or $Z$.|
|All $YZ$ is either $X$ or $W$.|
What conclusion can be drawn?
(a) Which regions in Figure 1 should be shaded to convey the statement "Every $Y$ is either $X$ and not $Z$, or $Z$ and not $X$"?
(b) Which regions in Figure 1 should be shaded to convey the statement "Every $WY$ is either both $X$ and $Z$ or neither of the two."?
(c) Which regions in Figure 1 should be shaded to convey the statement "All $XY$ is either $W$ or $Z$."?
(d) Which regions in Figure 1 should be shaded to convey the statement "All $YZ$ is either $X$ or $W$."?
(e) Solve Venn's problem by first shading the regions in Venn's four-set diagram in Figure 1 and then determining the appropriate conclusion.
2. Each of the four statements in Venn's problem in Exercise 1 can be translated into the language of sets using more modern notation. For example, "Every $Y$ is either $X$ and not $Z$ or $Z$ and not $X$ is equivalent to $Y\subset XZ'\cup ZX'$. Write the other three statements using set notation.
3. Venn's problem given in Exercise 1 can now be restated as a theorem. Fill in the first four blanks below using your answers to Exercise 2. Then fill in the last blank by translating your conclusion from Exercise 1 to set notation.
Theorem Let $X$, $Y$, $Z$, and $W$ be sets, and suppose
Now give a modern proof of this theorem.