1. Notions of Set Theory
Artin introduces a concept that is referred to as the canonical factoring of a map (function). The basic idea is that any function can be factored into three functions and in a somewhat unique way:
is onto, is a bijection, and is an injection. The construction of these three functions is done in a canonical, or natural, way that doesn’t require the use of objects outside the domain and/or range of .
Let be some non-empty set. If is a function from into a set , then we write
Suppose and . Then, we can form a composite function defined by for all . The associative law holds trivially for composition of functions.
Further, if , then the set of all the images of elements of , denoted by , is called the image of . In general, . We call the function onto whenever .
Now, let us partition the set into equivalence classes such that are in the same equivalence class iff . This partition is called the quotient set and is denoted by .
To illustrate, suppose and . Also, let such that and . Then, the quotient set, .
We construct now a function that maps each to its equivalence class. It can be verified that is onto. So, taking the above example, we have , , and .
Next, we construct a function where each element (which is an equivalence class) of is mapped to a where each is the image of the members of the equivalence class. Recall that are in the same equivalence class iff . Therefore, is one-to-one and onto. Continuing with our above example, we have , and .
And, finally, we construct a trivial function where for each . Note that is not an identity because it maps a subset, , into a possibly larger set, , i.e. is an identity iff is onto. In general, is one-to-one and into (i.e. an injection.)
Thus, if , we note that for every .
Once again, is onto, is a bijection, and is an injection.
It looks like it doesn’t make much sense to factor the way we did above, but we will explore more of this with respect to group homomorphisms in my next post.