O Sweet Mr Math

wherein is detailed Matt's experiences as he tries to figure out what to do with his life. Right now, that means lots of thinking about math.

Thursday, February 23, 2012

6:00 PM

Inherent in the idea of functions on sets, discussed last time, is collections of sets. If f(A)=B is a function defining the image B of a set A, where A is a subset of some set X, the domain of f isn't really X. The domain of f is really the collection of all subsets of X. This collection is called the power set of X, and it includes both the null set and X itself.

Before getting too involved with power sets, it's useful to develop some abstractions around collections of sets in general. Starting with a collection of sets, we often want to speak of a particular set in the collection. To do this, it's convenient to have an index. Essentially, we're defining a function that maps from one member of a set of indexes to one set in a collection of sets.

The term index could imply that the indexes are well organized and well behaved, but that is not necessarily the case. If the index set is the natural numbers, it's easy to start at the beginning and run through the entire set of indexes, even though it's infinite. However, use the real numbers as indexes to sets of rational numbers. For each real number, let the set be the rational numbers less than that real number. The real numbers are a set of indexes which map to sets that are part of a collection, but there is not an obvious way to list out the real numbers in a way that guarantees we will list out every one.

The indexes don't even necessarily have to be numbers. What we care about is that an index set exists and every member of the index set corresponds to one member of the collection of sets we are interested in, and that every set in the collection has an index in the index set.

We can use the index set to refer to intersections or unions of sets. An intersection of two sets is the set of points that are in both sets. A union of two sets is the set of points which is in either set. Extending this to indexed collection of sets, an intersection of indexed sets is the set of points that are in each set in the indexed collection. A union of indexed sets is the set of points that are in any set in the indexed collection.

This is a good time to mention complements of sets. If A is a subset of X, then the complement of A, Ac, (the notation for this varies) is all the points in X which are not in A. DeMorgan's laws state that the union of the complements of sets is equal to the complement of the intersection of the sets, and the intersection of the complements of sets is equal to the complement of the union. Symbolically, ∪(Xc) = (∩X)c and ∩(Xc) = (∪X)c. DeMorgan's laws for sets are the same as DeMorgan's laws for Boolean logic.

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