It's time for the Friday Random Ten!
- Led Zeppelin - I Can't Quit You Baby/How Many More Times
- Orianthi - Drive Away
- Coro Della Radio Svizzera - Ludwig van Beethoven: Der Glorreiche Augenblick, Op. 136, "Europa Steht!"
- Brave Combo - Kiss Me, Sweetheart
- Netherlands Bach Collegium - Johann Sebastian Bach: Cantata #61, "Komm, Jesu, Komm Zu Deiner Kirche"
- Alirio Diaz - Isaac Albeniz: Granada
- La Stravaganza Köln - Johann Sebastian Bach: Orchestral Suite #3, Gigue
- Netherlands Bach Collegium - Johann Sebastian Bach: Cantata #39, "Selig Sind, Die Aus Erbarmen"
- Staatskapelle Dresden - Ludwig van Beethoven: Leonore, "Mir Ist So Wunderbar"
- Kronos Quartet - Kuat Shildebaev: Kara Kemir
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.
Last time we talked about functions as mappings of points from one set to another set. The function maps from the domain to the codomain, but there may be points in the codomain that are not part of the mapping. The subset of the codomain that includes all the points in the mapping is the range of the function.
A function is one-to-one, or an injection, if every point in the range is mapped to from exactly one point in the domain. In other words, if x and y are points in the domain of the function f, f(x)=f(y) only if x=y. A function is onto, or a surjection, if the range of the function is equal to the codomain of the function. If the function is both one-to-one and onto, it is a bijection.
We can use the function to go from the domain to the range. It would be nice if we could go in the other direction, and nicer if going the other direction is also a function. If the original function is one-to-one, we can have an inverse function that maps from the range of the original function to the domain of the original function. If it's not one-to-one, then the inverse function would have points in its domain that map to multiple points in its range, so it wouldn't be a valid function.
We can also have functions act on sets of points rather than individual points. If a function f has domain X and codomain Y, then any point x∈X maps to a point in Y. If A is a subset of X, then we can find the set of points in Y that all the points in A map to. Instead of saying f(x)=y, where x and y are points in X and Y, we can say that f(A)=B, where A and B are subsets of X and Y.
The nifty thing about this is that the one-to-one and onto concepts apply to functions as sets the same way that they apply to functions of points. A function that maps sets is one-to-one if no subset of the codomain can be mapped to from more than one subset of the domain. If A and B are subsets of the codomain, then f(A)=f(B) only if A=B for one-to-one functions. And a function is onto if every subset of the codomain is the mapping of a subset of the domain.
The convenient part of this is that if a function is one-to-one or onto for points, it is also one-to-one or onto for sets. Also, looking at sets, every function has an inverse function. The inverse mapping can go from a single point to multiple points, but it can't go from a single set to multiple sets.
If A is set in the domain and B=f(A) is a set in the codomain, then B is the image of A. If f−1 is the inverse of f, then f−1(B)=A, and A is the preimage of B.
Now I think I've covered the real numbers in enough detail to move on to the next topic, which is functions and sets. Intuition about functions from high school algebra is a good place to start, but as usual, we want to generalize the concept and examine it in more detail.
A function, broadly speaking, is a mapping from members of one set to members of another set. Strictly speaking, the members of the sets don't have to be numbers, although in practice they often are, and the function can be written as a mathematical expression. The set that we are mapping from is the domain of the function. The important rule about functions and domains is that every member, or point, of the domain set maps to exactly one point in the set the function is mapping to, or the codomain.
There is no similar restriction on the codomain. Points in the codomain may not be reachable from any point in the domain. A simple example makes this clear. Define a function f as a mapping from the real numbers to the real numbers. We can label the domain as X and the codomain as Y, to be clear that the two sets are distinct, although they both have as members all real numbers. The notation for this is f:X→Y.
We will define f as a formula that maps points in domain into the codomain. For any point x∈X, f(x)=x2. For any x in the real numbers (the domain), we can find an f(x) in the codomain (also the real numbers), and no x in the domain maps to more that one point in codomain. However, there are points in the codomain that cannot be reached by this mapping, since no real number squared is a negative number. The subset of the codomain of points that can be mapped to from the domain is called the range. For this function, the range is the set of non-negative real numbers.
If the range of a function is equal to the codomain of the function, we can say that the function is onto or that it is a surjection.
The other thing to notice about f(x)=x2 is that for any positive f(x), there are two different numbers in the domain which map to that number. If every point in the range is mapped to by only one point in the domain, then the function is one-to-one or is an injection.
Any given function can be one-to-one, onto, both, or neither. If a function is both one-to-one and onto, it is a bijection.
For whatever reason, I am personally more comfortable with the terms injection, surjection, and bijection than I am with one-to-one and onto. I think "one-to-one" implies bijection to me, which is not correct. My goal is to deliberately use the words one-to-one and onto here to try to train myself to use them correctly.
Before I got distracted by complex numbers, I had been talking about euclidean spaces. I left off after introducing the norm, or length, of a vector in euclidean space. As a continuation of that topic, I want to look at the properties of norms in euclidean spaces. Specifically, if x and y are vectors in a euclidean space, let's look at ||x−y||, or the distance from y to x.
||x−y|| is always greater than or equal to 0. In fact, it's only equal to 0 if x=y. Since x−y is a vector, it is clear that the distance is always positive because the norm is calculated as a square root, which is a positive number.
||x−y|| = ||y−x||. The norm is a square root of a dot product of a vector with itself. The dot product is a sum of products, and since both vectors are the same, it becomes a sum of squares. Since a2=(−a)2, the order of the subtraction of x and y does not matter.
||x−y||≤||x−z||+||z−y||. This follows from the triangle inequality, as mentioned in my previous posts about euclidean spaces.
These three properties were written in terms of norms, but they could be rewritten as distances. And just like fields as a general concept were defined from the properties of addition and multiplication for the rational numbers, we can define metric spaces based on these three properties for distance. These three properties define a distance function, and any set of points with a defined distance function that has these properties for every point in the set is a metric space.
This is where I get nervous. I've studied abstract algebra enough to have some familiarity with fields other than the real numbers or complex numbers, so I have some sense of the distinction between the two, and I feel like I can recognize if something is true for the real numbers but might not be for fields in general. In addition, my impression so far is that real analysis isn't dependent on fields as much as it is on the real numbers, so confusion on the differences is not likely to be a problem for studying real analysis.
I have not studied topology, and metric spaces are (I think) a topology concept. It's also not clear to me how dependent real analysis is on metric spaces. I am somewhat concerned that I will take my one example of metric spaces (Euclidean k-spaces with distance defined as the square root of the dot product) and I will go off and draw all sorts of invalid conclusions about metric spaces in general. But this is what the textbook and course notes are giving me to work with, so I will press on and hope against catastrophic errors.