E-320: Teaching Math with a Historical Perspective Oliver Knill, 2010Lecture 7: The Boolean ring of sets1. ObjectiveStarting with the algebraic operations + and · on the set of all subsets, we derive unio ns, differ-ences or complements. We use the graphical pictures of Venn diagrams to see graphically whyassociativity and distributivity holds in the ring of sets.2. The Boolean ringTwo sets A, B of X.AdditionA + B = A∆B0 = ∅MultiplicationA · B = A ∩ B1 = X.1) How can we write the union of two sets A ∪ B using addition and multiplication.2) Verify that A = −A. The set A is the unique set which satisfies A + A = ∅.3) Draw the Venn diagr am of A − B = A + (−B).4) How can we write the set difference A \ B using addition and multiplication. (Note that thisis not the difference A − B.5) Draw a diagram which illustrates the associativity property A + (B + C) = (A + B) · A + C foraddition.6) Draw a diagr am which illustrates the associativity property A · (B · C) = (A · B) · C for multi-plication.7) Draw a diagra m which illustrates the distributive property A · (B + C) = A · B + A · C.8) Which sets have an inverse A−1in the sense that A · A−1= 1 = X?E-320: Teaching Math with a Historical Perspective Oliver Knill, 2010Lecture 7: Additional structure on sets1. ObjectiveWe look at two examples of structures which are using heavily the language of set theory. One ofthem is topology, an other is measure theory which is the foundation of probability theory. Thetwo definitions look very similar. But the structures which are defined are very different. Theyare in different areas of mathematics.1. The two structuresA set O of subsets of X is called a topology, ifT1) for any subset A of O the union of all sets A in A is in O.T2) for A, B ∈ O the intersection A · B is in OT3) the entire set X and the empty set are both in OElements in O are called open sets used as ”neighborhoods”.A set O of subsets of X is called a σ-algebra, ifS1) for any countable subset A, the union of all sets A in A is in O.S2) for A, B ∈ O the intersection A · B and addition A + B is in OS3) the entire set X and the empty set are both in O.Elements in O are called measurable sets and are used as ”events”.Remarks: The two structures a r e very important in mathematics. With a topological struc-ture one can define what is ”near” even if we have not a distance o n X. Many topological spacescan be realized as metric spaces with a distance function d(x, y), but there are some which are not.With suitable σ-algebra structures, one can define probabilities P which satisfy P [SnAn] =PnP [An] for disjoint events An, even if one can not count on X. This construction providesa solutio n to the Banach-Tarski paradox: The set of all subsets on the unit ball is not a goodsigma algebra. One needs a smaller one in order to have a proba bility which corresponds tovolume.Problem 1) : can you point out differences between these two structures? Especially, is a topologynecessarily a Boolean ring?Problem 2): check that the set of all subsets is both a topology as well as a σ -algebra.Problem 3): if Ω is a finite set. Is any topology also a σ-algebra?Problem 4): If Ω = {a, b, c}. How many topologies can you find on this set?Problem 5): How many different σ-algebras can you build on this set {a, b, c} .E-320: Teaching Math with a Historical Perspective Oliver Knill, 2010Lecture 7: Paradoxa1. ObjectiveWe look at examples of paradoxical statements which are at the heart of the incompletenesstheorems.2. The Russell paradoxThe set of all sets which do not contain themselves a s elements.a) Verify that if X contains itself as an element then X contains itself as an element.b) Verify t hat if X does not contain itself as an element then X does not contain itself as anelement.3. The barber paradoxA barber in Cambridge only shaves people who do not shave them-selves. Does the barber shave himself?c) There is an elegant (non mathematical) solution t o this para dox. Can you find it?4. The liars paradoxThe first version of the Liar Paradox seems have been found in the fourth century BC by Greekphilosopher Eubulides, successor of Euclid. Somebody tells ”I’m always lying”. Does the persontell the truth? The paradox is attributed to the Cretan philosopher Epimenides who said ”AllCretans are liars.”The Cretean philopher Epimedes tells that all Cretans are liars.Here is a variant of this paradox. Can you complete it?”The next sentence is .......... The previous sentence is ............”E-320: Teaching Math with a Historical Perspective Oliver Knill, 2010Lecture 7: Cardinality1. ObjectiveWe explore some cardinality questions. We look at a continuous map from the unit interval ontothe unit square. And explore whether it is a bijection.1. Cardinality of fractionsProblem 1) You might have read that in Eves but how can we find a bijection between the naturalnumbers and all positive fr actions? Even if you should not have read it, you might be able to getthe idea. Arrange the fractions in a suitable way so that you can count through them nicely:2. Peano curves in the planeThe following Peano curve construction has been devised by David Hilbert.first stage second stage third stage fourth stage2) The map f between the interval I = [0, 1] and the square Q is constructed recursively. Canyou see what is f[1/2]?3) Does the Peano map prove that the cardinality of the reals and the cardinality of the plane isthe same?By the way, the Peano construction works also in higher dimensions. Here are Peano curves inspace providing a continuous map from the interval onto the unit cube.first stage second stage third stage fourth