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**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:

, where

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 .

And, so,

.

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.

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