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# 4: Sets

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No more turkey, but I’d like some more of the bread it ate.

–Hank Ketcham

• 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.
• 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.
• 4.3: Set Operations
In this section, we’ll continue to develop the correspondence between Logic and Set theory.
• 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.