4: Sets
No more turkey, but I’d like some more of the bread it ate.
–Hank Ketcham
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- 4.1: Basic Notions of Set Theory
- In modern mathematics, there is an area called Category theory which studies the relationships between different areas of mathematics. More precisely, the founders of category theory noticed that essentially the same theorems and proofs could be found in many different mathematical fields. In this sort of situation, one can make what is known as a categorical argument in which one proves the desired result in the abstract, without reference to the details of any particular field.
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- 4.2: Containment
- There are two notions of being “inside” a set. A thing may be an element of a set, or may be contained as a subset. Distinguishing these two notions of inclusion is essential. One difficulty that sometimes complicates things is that a set may contain other sets as elements. For instance, as we saw in the previous section, the elements of a power set are themselves sets.
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- 4.4: Venn Diagrams
- Hopefully, you’ve seen Venn diagrams before, but possibly you haven’t thought deeply about them. Venn diagrams take advantage of an obvious but important property of closed curves drawn in the plane. They divide the points in the plane into two sets, those that are inside the curve and those that are outside! (Forget for a moment about the points that are on the curve.) This seemingly obvious statement is known as the Jordan curve theorem, and actually requires some details.
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- 4.5: 4.5 Russell’s Paradox
- Bertrand Russell was one of the twentieth century’s most colorful intellectuals. He was perhaps better known as a philosopher. It’s hard to conceive of anyone who would characterize Russell as an applied mathematician! In the beginning of our investigations into Set theory, we mentioned that the notion of a “set of all sets” leads to something paradoxical. Now we’re ready to look more closely into that remark and hopefully gain an understanding of Russell’s paradox.