The direct sum of functions

In this post,I will talk about the direct sum of two functions.I will divide this post into five parts.This is the first part.And below,the second and the third part will show you the definition.The fourth part will show you the proof.The last part will show you the “insights” of the direct sum of functions,so that is the most important part.

Given sets $X,Y$ and their cartesian product $X\times Y$ .Now let’s define two maps : $\phi :X\times Y\to X$ and $\psi :X\times Y\to Y$ .The rule of these two maps are given below:

$(x,y)\underrightarrow{\phi} x$

$(x,y)\underrightarrow{\psi} y$ .

Given a set $Z$ .And two functions $f:Z\to X$ , $g:Z\to Y$ .We define the direct sum of $f$ and $g$ as a function $h:Z\to X\times Y$ ,where $h$ satisfies the following property:

$\phi\circ h=f$ , $\psi\circ h=g$ . We denote $h$ as $x\bigoplus y$ .

Now that $h$ has been defined,we need to prove its existence and uniqueness.I do not want to discuss the rigorous proof here.On the one hand,the rigourous proof here is easy,on the other hand, mathematics is more than rigorous proofs.Usually,long time after learning a material,what leaves in your head are not rigourous proofs,but some sort of intuitive graphs.So I’d like to talk about the direct sum intuitively(I think this intuitive proof is very near to the rigorous one):

For any fixed element $z\in Z$ , $f(z)$ is a fixed element in $X$ , $g(z)$ is a fixed element in $Y$ .So $(f(z),g(z))$ is a fixed element in $X\times Y$ .So $h$ exists: $h:z\to (f(z),g(z))$ .And,because $(f(z),g(z))$ is fixed,so for any fixed $z$ ,there is no other choice,so $h$ is unique. $\Box$

Now it is the time to show the “insight” of the definition of the direct sum.(My English is not good,but I will try my best to explain it well )We know that the cartesian product $X\times Y$ consists of two ingredients: $X$ and $Y$ .In fact, $X$ and $Y$ are irrelevant.(By way of analogy,we can regard $X$ and $Y$ as two person,they live in different places,they don’t know each other.)A map from $Z$ to $X\times Y$ form a connection between $Z$ and $X\times Y$ .We can see the connection between $Z$ and $X\times Y$ separately,that is ,

the connection between $Z$ and $X\times Y$ =the connection between $Z$ and $X$ together with the connection between $Z$ and $Y$ .This is the spirit of direct sum.In fact, I think “direct” is a very suitable word to characterize such property.