Venn diagrams and prime numbers
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The search for rotationally symmetric Venn diagrams made up of ellipses has led to some elegant work. Mathematicians had already run into difficulties with four ellipses when David Henderson proved in 1963 that no such diagrams existed for non-prime n! Under rotation, each region representing the intersection of k sets belongs to a rotational orbit of n such regions. Moreover, there are regions representing the intersection of k sets, so n must always divide .
For example, when n = 4, there must be six regions representing the intersection of two sets, but 4 does not divide 6...
It was not until 1975 that Yugoslav mathematician Branko Grünbaum produced an example of a symmetric Venn diagram with five ellipses. In 2002, Peter Hamburger produced an example of a symmetric Venn diagram with eleven ellipses; a year later, Jerrold Griggs, Charles Killian and Carla Savage finally proved that such a diagram can indeed be drawn for every prime number n.
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Axioms—and rules of logic too!
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Set theory is an axiomatic theory: we begin with statements that we accept as true and from which we then deduce theorems. The axioms specific to set theory are supplemented by the axioms and rules of inference of classical logic. Among these rules, the following statement seems self-evident but must be stated clearly: what is true of an arbitrary element of a set is true of every element of that set.
Modus ponens and modus tollens are two other basic rules of inference. Modus ponens, or detachment, states that from p and p implies q, we may infer q. For example, let p mean "it is raining" and let p implies q mean "if it is raining, then I take out my umbrella." Applying modus ponens on a rainy day, we may infer that I have my umbrella. Modus tollens, meanwhile, is a rule of inference stating that a proposition and its contrapositive are logically equivalent. For example, if the proposition is "if it is raining, then I take out my umbrella," and I happen not to have my umbrella, modus tollens allows us to infer that it is not raining.
By combining axioms using rules of inference, we build theorems. These are nothing more than syntactic consequences of the original axioms!
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Cantor: two proofs of a theorem
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Georg Cantor became interested in infinite sets in the early 1870s. One question he explored was whether infinite sets of different cardinalities existed. In particular, was there a bijection between the natural numbers and the real numbers, for which he had just given a construction? Cantor answered this question in the negative as early as 1874. His rather cumbersome proof used a nested-interval argument together with the fact that every bounded increasing sequence converges.
Eighteen years later, he returned to the subject while developing his theory of infinite cardinals. He then gave a new, much simpler proof. An easy extension of it also shows that the power set of any set has cardinality strictly greater than that of the set itself. This result is known as Cantor's theorem. A slightly modified form of his second proof was popularized as Cantor's diagonal argument.
