A function is a binary relation satisfying certain conditions, and we have already defined the composition and inverse of binary relations. For the composition and inverse of functions to be well-defined, the result of composing them as binary relations or taking their inverse as binary relations must be a function.
Composition of Functions
The composition of functions is straightforward.
Proposition 1 Consider functions \(f:A\rightarrow B\) and \(g:B\rightarrow C\). Then \(g\circ f\) is a function from \(A\) to \(C\).
Proof
It is clear that the domain of \(g\circ f\) is all of \(A\). This is because the value of \(f\) is defined for all elements of \(A\), and the value of \(g\) is defined for all elements of \(B\), in particular for all elements of \(f(A)\subseteq B\). Therefore, to prove the given proposition, it suffices to show:
For any \(x\in A\), if \((x,z)\) and \((x,z')\in G\circ H\), then necessarily \(z=z'\).
Suppose \((x,z),(x,z')\in G\circ F\). By the definition of \(G\circ F\), there exist \(y\) and \(y'\) such that \((x,y)\in F\), \((y,z)\in G\), and \((x,y')\in F\), \((y',z')\in G\), respectively. Since \(f\) is a function, from \((x,y)\in F\) and \((x,y')\in F\), we have \(y=y'\). Now from the conditions \((y,z)\in G\) and \((y',z')\in G\), together with \(y=y'\) and the fact that \(g\) is a function, we conclude \(z=z'\).
Therefore, the composition of functions is nothing more than the composition of binary relations, and the resulting binary relation is always a function.
Inverse Functions
Defining inverse functions is somewhat more involved. While we can consider the inverse relation \(f^{-1}\) of a function \(f\) as a binary relation, this relation may not be a function. To determine when \(f^{-1}\) is a function, we must first define injective, surjective, and bijective functions.
Definition 2 Consider a function \(f:A\rightarrow B\). The function \(f\) is injective if any two distinct elements of \(A\) have distinct values under \(f\). The function \(f\) is surjective if \(f(A)=B\). If \(f\) is both injective and surjective, then this function is said to be bijective.
These terms have only recently become standard in the broader history of mathematics. Previously,
- Instead of injection, the term one-to-one was used,
- Instead of surjection, the term onto was used,
- Instead of bijection, the term one-to-one and onto was used,
and the Korean terms 일대일함수 (one-to-one function) and 일대일대응 (one-to-one correspondence) commonly used in high school are remnants of this earlier terminology.
Example 3 Let \(A\subseteq B\). If we define \(f:A\rightarrow B\) by \(x\mapsto x\), this function is injective. This function is called the canonical injection.
For any function \(f:A\rightarrow B\), subset \(X\subseteq A\), and canonical injection \(i:X\hookrightarrow A\), it is easy to verify that
\[f|_X=f\circ i\]holds. Sometimes the right-hand side is written as \(i_\ast f\).
Example 4 By definition, it is clear that \(\id_A=(\Delta_A,A,A)\) is bijective.
Now we can define the inverse function as promised.
Proposition 5 For a function \(f:A\rightarrow B\), \(f^{-1}\) is a function if and only if \(f\) is bijective.
Proof
If \(f^{-1}\) is bijective, then it is also a surjective function, so its domain becomes \(B\). Also, since \(f\) is injective, \(f^{-1}\) becomes a function.
Conversely, suppose \(f^{-1}\) is a function. Then by definition, \(\pr_1 f^{-1}=B\). However, substituting \(R_2=\id_A\) and \(R_1=f^{-1}\) into the first equation of §Operations on Binary Relations, ⁋Proposition 8 gives \(\pr_1f^{-1}=f(A)\), so \(B=f(A)\), and therefore \(f\) is a surjective function.
Also, suppose \((x,f(x))\in F\) and \((y, f(y))\in F\) are well-defined. Then \((f(x), x)\in F^{-1}\) and \((f(y),y)\in F^{-1}\). Furthermore, if \(f(x)=f(y)\), then from the fact that \(f^{-1}\) is a function, we have \(x=y\). Therefore, \(f\) is an injective function.
This defined \(f^{-1}\) is called the inverse function of \(f\). We can easily verify that \(f^{-1}\circ f=\id_A\) and \(f\circ f^{-1}=\id_B\).
The following remark provides important intuition for defining retractions and sections in the next article.
Remark The two equations \(f^{-1}\circ f=\id_A\) and \(f\circ f^{-1}=\id_B\) remain partially true even if \(f\) is not bijective, but only surjective or injective.
- If \(f\) is injective, then \(f\) is a bijection between \(A\) and \(f(A)\subseteq B\), so \(\tilde{f}^{-1}:f(A)\rightarrow A\) exists. Now \(\tilde{f}^{-1}\circ f=\id_A\).
- If \(f\) is surjective, then for every \(y\in B\), there always exists some \(x\) such that \(f(x)=y\). Letting \(\tilde{f}^{-1}\) be the function that maps each such \(y\) to \(x\), we have \(f\circ \tilde{f}^{-1}=\id_B\).
Product of Functions
Definition 6 A function of two variables is a function whose domain is a set of ordered pairs.
If \(f\) is a function of two variables, we write \(f(x,y)\) instead of \(f((x,y))\) to denote the value of \(f\) at \((x,y)\). The most significant feature of such functions is that we have two variables to manipulate when observing the behavior of \(f\). For example, to see how \(f\) changes as the first coordinate varies, we can fix the second coordinate and treat \(f\) as a function taking only the first coordinate as input.
Definition 7 Let \(f:A\rightarrow B\) be a function of two variables. For every \(y\), define \(A_y\) as the set of all \(x\) such that \((x,y)\in A\). Then the function \(x\mapsto f(x,y_0)\) from \(A_{y_0}\) to \(B\) is called the partial mapping of \(f\) at \(y_0\), and is written as \(f(-,y_0)\). Similarly, \(f(x_0,-)\) is also defined.
For any two functions \(u:A\rightarrow C\) and \(v:B\rightarrow D\), we can always combine them to create a function of two variables from \(A\times B\) to \(C\times D\). That is, we define
\[z\mapsto (u(\pr_1 z),v(\pr_2z))\]This function is called the product of \(u\) and \(v\), and is written as \(u\times v\). Of course, this function has nothing to do with a function created by multiplying the two values \(u(x)\) and \(v(x)\).
References
[Bou] N. Bourbaki, Theory of Sets. Elements of mathematics. Springer Berlin-Heidelberg, 2013.
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