Andrew Cooper

Recall that we defined inverses for any relation:

Definition. If $R$ is a relation from $A$ to $B$, then $R^{-1}$ is a relation from $B$ to $A$, defined by $(b,a)\in R^{-1}$ iff $(a,b)\in R$.

Thus (in our formalism) every function has an inverse. The inverse of a function so defined, however, may not be a function itself! (Remember that most operations we do to sets or to relations will not preserve functionhood.) So let’s discover when taking the inverse (“swapping $x$ and $y$”) actually does result in a function.

Recall that $g:A\rightarrow B$ means that two conditions hold:

1. $\operatorname{Domain}(g)=A$
2. $\left((a,b)\in g\text{ and }(a,c)\in g\right)\Rightarrow b=c$.

Exercise 1. Suppose $f:X\rightarrow Y$. By substituting $f^{-1}$ for $g$, $X$ for $A$, and $Y$ for $B$, write what it means for $f^{-1}:Y\rightarrow X$.

Exercise 2. Use the definition of $f^{-1}$ to rewrite your conditions solely in terms of $f$ (with no $f^{-1}$ anywhere).

Definition.  Let $f:X\rightarrow Y$. We call $f$ onto or surjective if $\forall y\in Y, \exists x\in X: y=f(x)$. We call $f$ one-to-one or injective if for any $x,z\in X$, we have that $f(x)=f(z)$ guarantees $x=z$.

Definition. A function which is both one-to-one and onto is called a one-to-one correspondence or a bijection.

The following theorem is important, but its proof is just the combination of Exercises 1 and 2.

Theorem. Let $f:X\rightarrow Y$. Then $f^{-1}:Y\rightarrow X$ iff $f$ is both one-to-one and onto.

Theorem.  Suppose $f:X\rightarrow Y$ is one-to-one and $g:\operatorname{Range}(f)\rightarrow Z$ is one-to-one. Then $g\circ f:X\rightarrow Z$ is one-to-one.

Proof. Note that the condition that $g$ be defined on the range of $f$ is necessary in order to compose the two functions as functions.

Let $x,z\in X$. Suppose $g\circ f(x)=g\circ f(z)$. Applying the fact that the composition of functions is a function, we have \begin{align*}g(f(x))=g(f(z))\end{align*}

Since $g$ is one-to-one, we have $f(x)=f(z)$. Since $f$ is one-to-one, we have $x=z$. Since $x,z$ were arbitrary, we’ve shown $g\circ f$ is one-to-one. $\Box$.

Theorem. Suppose $f:X\rightarrow Y$ is onto and $g:Y\rightarrow Z$ is onto. Then $g\circ f$ is onto.

Proof. This is an exercise for you.

Consider the set of all functions $f:X\rightarrow \{0,1\}$. Each subset $A\subseteq X$ has such a function, namely its characteristic function, $\chi_A$; moreover, given $f:X\rightarrow\{0,1\}$ we can define the set $A=f^*(\{1\})$ and see that $f=\chi_A$. So the functions $f:X\rightarrow \{0,1\}$ correspond to subsets of $X$.

Definition. Given sets $A$, $B$, define \begin{equation*}A^B=\left\{f:B\rightarrow A\right\}\end{equation*} to be the set of all functions from $B$ to $A$.

This notation connects with our powerset notation by writing $2$ for the set $\{0,1\}$ (which has two elements).

Theorem. If $A$ has exactly $n$ elements and $B$ has exactly $m$ elements, then $A^B$ has exactly $n^m$ elements.

Proof. To define a function $B\rightarrow A$ requires choosing, for each $b\in B$, one of the $n$ elements of $A$.

Each time we make such a choice, there are $n$ options; and we have to make the choice $m$ times. So in total there are $n\cdot n\cdot\cdots \cdot n=n^m$ possibilities. $\Box$

[callout headingicon=”noicon” textalign=”textleft” type=”basic”]All in all, those who were chained would consider nothing besides the shadows of the artifacts.

Plato, The Allegory of the Cave

[/callout]

Pushforwards and Pullbacks

A function $f:X\rightarrow Y$ gives us a way of taking elements in $X$ and pushing them over to $Y$. We call $f(x)$ the image of $x$ under $f$. In fact, we can also think of $f$ as giving a way to push a set of elements of $X$ over to $Y$.

Definition. Given $f:X\rightarrow Y$ and $A\subseteq X$, we define $f_*A=\left\{f(x)\middle\vert x\in A\right\}$. We call $f_*A$ the image of $A$ under $f$ or the pushforward of $A$ by $f$.

Notice that by definition, $f_*A$ is a subset of the target $Y$.

E.g. If $f:t\mapsto t^2$, and $A=[-2,3]$, then $f_*A=[0,9]$.

Proof.  ($\Rightarrow$) Let $y\in f_*A$. Then $\exists x\in A: f(x)=y$. So $-2\leq x\leq 3$.

If $-2\leq x\leq 0$, then $y=x^2$ is between $0$ and $4$. So $y\in[0,9]$.

If $0\leq x\leq 3$, then $y=x^2$ is between $0$ and $9$. So $y\in[0,9]$.

($\Leftarrow$) Let $y\in[0,9]$. Then $0\leq y\leq 9$. Let $x=\sqrt{y}$. So $0\leq x\leq 3$. Then $f(x)=\left(\sqrt{y}\right)^2=y$. Since $0\leq x\leq 3$, $x\in[-2,3]$.

Notation. Most sources write $f(A)$ for the image of $A$ under $f$. This is not an entirely unreasonable choice, but it means that the symbol $f(A)$ has two interpretations — if $A$ is an element of $X$, then $f(A)$ is an element of $Y$; if $A$ is a subset of $X$, then $f(A)$ is a subset of $Y$. Our notation $f_*A$ is always going to be a set. But you should know, for your future mathematics courses, that $f(A)$ is a standard notation.

Theorem/Realization. If $f:X\rightarrow Y$, then $f_*:2^X\rightarrow 2^Y$.

That is, $f_*$ takes subsets of $X$ and yields subsets of $Y$.

Definition. Given $f:X\rightarrow Y$ and $B\subseteq Y$, we define $f^*B=\left\{x\middle\vert f(x)\in B\right\}$. We call $f^*B$ the preimage of $B$ under $f$ or the pullback of $B$ by $f$.

Note that $f^*B$ is a subset of the domain $X$.

E.g. If $f:\mathbb{R}\rightarrow\mathbb{R}$ is given by $t\mapsto t^2$, then $f^*((-2,-1))=\varnothing$ and $f^*([-4,4))=[0,2)$.

Theorem/Realization. If $f:X\rightarrow Y$, then $f^*:2^Y\rightarrow 2^X$.

Notation. Just like with the image, there is a standard-but-terrible notation for the preimage. Most sources write $f^{-1}(B)$ for the preimage of $B$ under $f$. This is just nutty, because the object in question doesn’t have anything to do with inverses. Nevertheless, you should expect at least once in your mathematical life to have to deal with the string of symbols $f^{-1}(B)$, where $B$ is some set. We live in a fallen world.

Theorem. Let $f:X\rightarrow Y$, $A\subseteq X$, $B\subseteq X$, $C\subseteq Y$, $D\subseteq Y$. Then

1. $f_*(A\cup B)=f_*(A)\cup f_*(B)$
2. $f_*(A\cap B)\subseteq f_*(A)\cap f_*(B)$
3. $f^*(C\cup D)=f^*(C)\cup f^*(D)$
4. $f^*(C\cap D)=f^*(C)\cap f^*(D)$
5. $f^*(C^c)=\left(f^*(C)\right)^c$

Proof. This is an exercise, but it’s just an exercise in proving set equalities.

Characteristic Functions

Recall that for $S$ to be a set means that “$x\in S$” is an open sentence. We can view this in terms of functions.

Definition. Given a set $A$, the characteristic function of $A$ is the function \begin{equation*} \chi_A(x)=\begin{cases}1 &\text{ if } x\in A\\0&\text{ otherwise}\end{cases} \end{equation*}

Technically in order to define this function, we need to specify a universe for $x$ — but notice that it really doesn’t much matter, because as long as $x\notin A$, $\chi_A(x)=0$.

Exercise. If $A$ and $B$ are subsets of some common universe $U$, then for any $x\in U$, we have:

1. $\chi_{A\cap B}(x)=\chi_A(x)\chi_B(x)$,
2. $\chi_{A\cup B}(x)=\max\{\chi_A(x),\chi_B(x)\}$
3. $\chi_{A^c}(x)=1-\chi_{A}(x)$.

Theorem.  $\chi_A^*(\{1\})=A$.

This says that $A$ determines its characteristic function, and $\chi_A$ determines $A$ as well.

Set Operations on Functions

Functions are sets, so we can try to combine them like we combine sets. That is, we can ask the following questions:

• Is the complement of a function necessarily a function?
• Is the intersection of two functions necessarily a function?
• Is the union of two functions necessarily a function?
• Is the set product of two functions necessarily a function?

Let’s investigate the first three; the last one I’ll leave for homework.

To start with, let’s pick two functions and run with them. How about: $f:x\mapsto x+1$ and $g:t\mapsto t^2$.

The complement of $g$ is $g^c=\left\{(x,y)\middle\vert y\neq x^2\right\}$, shown here in hatching:That’s definitely not a function: it badly fails the vertical line test.

What about intersection?

The intersection is just two points; so $f\cap g$ is not a function from $\mathbb{R}$ to $\mathbb{R}$. But $f\cap g$ is a function from a smaller set.

Let’s look at $f\cup g$:

It also fails the vertical line test. But:

The Pasting Lemma (in the category of sets). Let $f:X\rightarrow Y$ and $g:A\rightarrow B$ be two functions. Then $f\cup g:X\cup A\rightarrow Y\cup B$ if and only if $f|_{A\cap X}=g|_{A\cap X}$.

Proof. This is a biconditional, so we have two directions:

($\Leftarrow$) Assume $f|_{A\cap X}=g|_{A\cap X}$. We need to show $f\cup g$ is a function; to do this, there are two clauses.

(clause 1: domain) Let $t\in X\cup A$. Then either $t\in X$ or $t\in A$.

If $t\in X$, then because $f:X\rightarrow Y$, $\exists y\in Y: (t,y)\in f$. Then $(t,y)\in f\cup g$.

If $t\in A$, then because $g:A\rightarrow B$, $\exists b\in Y: (t,b)\in g$. Then $(t,b)\in f\cup g$.

In either case, there’s an element of $z\in Y\cup B$ so that $(t,z)\in f\cup g$. That’s clause 1 of the definition of function.

(clause 2: unambiguousness) Let $t\in X\cup A$ and $z_1,z_2\in Y\cup B$ be such that $(t,z_1)\in f\cup g$ and $(t,z_2)\in f\cup g$. There are four cases:

If $(t,z_1)\in f$ and $(t,z_2)\in f$, then the fact that $f$ is a function implies $z_1=z_2$.

If $(t,z_1)\in g$ and $(t,z_2)\in g$, then the fact that $g$ is a function implies $z_1=z_2$.

If $(t,z_1)\in f$ and $(t,z_2)\in g$, then we have $t\in \operatorname{Domain}(f)=X$ and $t\in\operatorname{Domain}(g)=A$, so $t\in X\cap A$. But then $z_1=f(t)=f|_{X\cap A}(t)=g|_{X\cap A}(t)$. So $(t,z_1)\in g|_{X\cap A}\subseteq g$. So $(t,z_2)\in g$. Since $g$ is a function, $z_1=z_2$.

If $(t,z_1)\in g$ and $(t,z_2)\in f$, then we have $t\in \operatorname{Domain}(f)=X$ and $t\in\operatorname{Domain}(g)=A$, so $t\in X\cap A$. But then $z_2=f(t)=f|_{X\cap A}(t)=g|_{X\cap A}(t)$. So $(t,z_2)\in g|_{X\cap A}\subseteq g$. So $(t,z_1)\in g$. Since $g$ is a function, $z_1=z_2$.

In any of these four cases, $z_1=z_2$, which is what we needed for clause 2 of the definition of function.

($\Rightarrow$) In this direction, we need to show two functions are equal. So we should probably use the TAWFAE. I’ll leave this direction to you.

Why does this theorem bear such a silly name? Consider this diagram, which tells you how to build a cube from a piece of paper. In order to attach the sides, you have to paste the red tab to the back of one of the appropriate square (I’ve indicated where the red tab gets attached by coloring it green). We won’t succeed in pasting unless the red tab and the green target match up.

That’s what the condition $f|_{X\cap A}=g|_{X\cap A}$ ensures: that the functions $f$ and $g$ agree on the overlap of their domains.

Again, an example from precalculus mathematics: when we define a function piecewise, like $$h(x)=\begin{cases}x^2 & \text{ if } x\leq 3\\ -x+1&\text{ if } x>3\end{cases}$$we usually want to make sure that the two pieces have nonoverlapping domains. This corresponds, in the Theorem above, to the case that $X\cap A=\varnothing$. If we weren’t so careful, we might have a situation like this:$$h(x)=\begin{cases}x^2 & \text{ if } x\leq 3\\ -x+1&\text{ if } x>1\end{cases}$$where it’s ambiguous what $h(2)$ is:

Composition of Functions

Because functions are relations, we can consider their composites. We have the following nice theorem:

The Composition of Functions Is a Function. If $f:X\rightarrow Y$ and $g:Y\rightarrow Z$, then $g\circ f:X\rightarrow Z$.

Proof. There are two things to show:

(clause 1: domain) Let $x\in X$. Then, because $f$ is a function, $\exists y\in Y:(x,y)\in f$. Since $y\in Y$ and $g$ is a function, $\exists z\in Z:(y,z)\in g$. Then $(x,z)\in g\circ f$. $\Box$.

(clause 2: unambiguousness) Given $x\in X, z_1,_2\in Z$ so that $(x,z_1)\in g\circ f$ and $(x,z_2)\in g\circ f$. We need to show $z_1=z_2$.

Since $(x,z_1)\in g\circ f$, there is some $y_1\in Y$ with $(x,y_1)\in f$ and $(y_1,z_1)\in g$. We also have $y_2\in Y$ with $(x,y_2)\in f$ and $(y_2,z_2)\in g$.

Since $f$ is a function, $y_1=y_2$. So we have $(y_1,z_2)\in g$. But since $g$ is a function, $(y_1,z_2)\in g$ and $(y_1,z_1)\in g$ forces $z_1=z_2$. $\Box$.

Important Formula. If $f$ and $g$ are functions, so that $g\circ f$ is a function, then $(g\circ f)(x)=g(f(x))$.

Proof. What does it mean for $z=(g\circ f)(x)$? It means $(x,z)\in g\circ f$. But that means $\exists y: (x,y)\in f$ and $(y,z)\in g$. So $y=f(x)$ and $z=g(y)$. By substituting, we obtain $z=g(f(x))$. $\Box$

This formula explains the “backwards” looking choice of how we write compositions: even though $f$ come first, we write $g\circ f$. The only way out of this would be to switch our function notation to something like $(x)f$, so that the composite would be $((x)f)g$. (Some people do this, and it has a name: reverse Polish notation. But most normal people don’t do it.)

When are two functions equal? A function is a kind of relation, which means it’s a kind of set. So two functions being equal means, fundamentally, that each is a subset of the other. That’s inconvenient, and it doesn’t really jibe with our understanding of functions as encoding rules.

Theorem About When Functions Are Equal (TAWFAE). Let $f$ and $g$ be functions. Then $f=g$ if and only if:

1. $\operatorname{Domain}(f)=\operatorname{Domain}(g)$, and
2. $\forall x\in \operatorname{Domain}(f), f(x)=g(x)$.

Proof. This theorem sort of looks like it will be complicated to prove, but let’s keep our wits about us and rely on logic to help us out.

Globally, this is a biconditional proof so it has two directions:

($\Rightarrow$) Assume $f=g$. We need to prove two things:

(clause 1) This is the claim that two sets are equal, so we have to prove each is a subset of the other:

($\subseteq$) Let $t\in \operatorname{Domain}(f)$. Then $\exists y: (t,y)\in f$. Since $f=g$, this means $(t,y)\in g$. So $t\in\operatorname{Domain}(g)$.

($\supseteq$) Let $t\in \operatorname{Domain}(g)$. Then $\exists y: (t,y)\in g$. Since $f=g$, this means $(t,y)\in f$. So $t\in\operatorname{Domain}(f)$.

(clause 2) This is a universal claim, so we start the same way as always: Let $x\in \operatorname{Domain}(f)$. Then $\exists y: (x,y)\in f$. We can rewrite this as $y=f(x)$. On the other hand, we know from clause 1 that $\operatorname{Domain}(g)=\operatorname{Domain}(f)$, so $\exists z: (x,z)\in g$. We can rewrite this as $z=g(x)$. But since $g=f$, we actually have $(x,z)\in f$.

Since $(x,y)\in f$ and $(x,z)\in f$, and $f$ is a function, $y=z$.

So $f(x)=y=z=g(x)$.

($\Leftarrow$) Assume both clauses 1 and 2 hold. In this direction, we need to prove two sets are equal.

($\subseteq$) Let $Q\in f$. Then $Q=(x,y)$ for some $x$ and some $y$. $x\in\operatorname{Domain}(f)$, so by clause 1, $x\in\operatorname{Domain}(g)$. Then there’s some $z$ with $(x,z)\in g$. We can rewrite in function notation as $y=f(x)$ and $z=g(x)$. By clause 2, we know $f(x)=g(x)$, so $y=z$. That is, $Q=(x,y)\in g$.

($\supseteq$) This is so similar to $\subseteq$, I’ll let you work it out.

Since we proved both directions, we’re done. $\Box$.

This theorem gives us a new way to prove two functions are equal, should we ever need it: we establish clause 1 (“they have the same domain”) and clause 2 (“they have the same rule”).

Restrictions and Extensions

In middle and high school algebra courses, we often ask questions like “What’s the domain of $f(x)=\frac{sin(x)}{x^2-25}$?”. According to the TAWFAE, this is actually a dumb question: TAWFAE says in order to know what a function is, we need to know what the domain is. So a question like doesn’t actually contain enough information to be answerable.

What the question is really asking, though is: what’s the most natural, or maybe what’s the largest possible domain on which this rule makes sense?

As a middle-school student, I was very confused by questions like this one:

Find the domain of $f(x)=\frac{x^2-9}{x+3}$.

My answer would usually go like this: that function is really the same as $x-3$, the domain of which is all of $\mathbb{R}$. And I would always get that question wrong. Why?

The functions $\psi:t\mapsto \frac{t^2-9}{t+3}$ and $\varphi:x\mapsto x-3$ do indeed have the same rule, in the sense that, whenever $\psi(t)$ makes sense, we have

$$ \psi(t)=\frac{t^2-9}{t+3}=\frac{(t-3)(t+3)}{t+3}=t-3=\varphi(t)$$

But the natural domain of $\psi$ is $\mathbb{R}\setminus\{-3\}$ and the natural domain for $\varphi$ is $\mathbb{R}$; so that means the domains are different — hence by the TAWFAE, we know $\psi\neq\varphi$. Nevertheless, the two functions are related.

Definition. If $f:X\rightarrow Y$ and $A\subseteq X$, we define the restriction of $f$ to $A$ as

\begin{align*}f|_{A}:A&\rightarrow Y\\ a&\mapsto f(a)\end{align*}

That is, $f|_A$ has the same rule as $f$, just we only care about inputs that come from $A$.

Equipped with this notion, we can describe the relationship between $\psi$ and $\varphi$:

$$\psi=\varphi|_{\mathbb{R}\setminus\{-3\}}$$

Definition. If $f:X\rightarrow Y$, and $X\subseteq Z$, and there is $g:Z\rightarrow Y$ with $g|_X=f$, we call $g$ an extension of $f$.

Here’s an illustration: if we let $f$ denote the function displayed in green, $g$ denote the function displayed in blue, and $h$ the function displayed in red, then we have that:

• $f|_{[-2,\infty)}=g$
• $f|_{[3,\infty)}=h$
• $g|_{[3,\infty)}=h$
• $f$ is an extension of $g$, from $[-2,\infty)$ to all of $\mathbb{R}$
• $g$ is an extension of $h$, from $[3,\infty)$ to $[2,\infty)$

Definition. A function $f$ from a set $X$ to a set $Y$ is a relation which satisfies the following two conditions:

1. $\forall x\in X, \exists y\in Y: (x,y)\in f$, and
2. $\forall x\in X, y_1,y_2\in Y, \left[(x,y_1)\in f \wedge (x,y_2)\right]\in f\Rightarrow y_1=y_2$

That is, a function assigns to each (clause 1) element of $X$ an element of $Y$, and that assignment is unambiguous (clause 2). Since we could rewrite clause 1 as $\operatorname{Domain}(f)=X$ (why?), we’ll refer to clause 1 as the “domain clause” and clause 2 as the “unambiguousness clause”.

Notation. Instead of the mouthful $f$ is a function from $X$ to $Y$, we write the shorthand $f:X\rightarrow Y$.

We also have this familiar notation:

Notation. If $f$ is a function from $X$ to $Y$, and $(x,y)\in f$, we write $y=f(x)$.

Observe that if $(x,y_1)\in f$ and $(x,y_2)\in f$, then this notation reads as follows: $y_1=f(x)$ and $y_2=f(x)$, so we’d darn well better have $y_1=y_2$, or else something is very wrong.

There is another way to represent a function, by specifying its domain and its rule:

Notation. If $f$ is a function from $X$ to $Y$, we write $f:X\rightarrow Y$. To specify the rule that $f$ uses, we write \begin{align*}f:X&\rightarrow Y\\x&\mapsto f(x) \end{align*}

For example, we could denote the function which takes in a real number as input, and squares it by

\begin{align*} sq:\mathbb{R}&\rightarrow\mathbb{R}\\ t&\mapsto t^2 \end{align*}

Notice that the arrow between the sets is a different shape from the one between elements of the set. As usual we want to let go of our $x$s, so we could as well have written

\begin{align*} sq:\mathbb{R}&\rightarrow\mathbb{R}\\ z&\mapsto z^2 \end{align*}

or

\begin{align*} sq:\mathbb{R}&\rightarrow\mathbb{R}\\ Q&\mapsto Q^2 \end{align*}

This frees us from the tyranny of always writing $y=f(x)$.

[callout headingicon=”noicon” textalign=”textleft” type=”basic”]On the road to the City of Skepticism, I had to pass through the Valley of Ambiguity.

Adam Smith

[/callout]

What is a function? The notion of a function is fundamental to mathematics, but like many fundamental aspects of mathematics, its definition was not nailed down until relatively recently. Mathematicians were using functions without thinking all that hard about what functions were! Let’s rectify that.

The definition that I prefer to use is the following:

Definition. A function from the set $X$ to the set $Y$ is an unambiguous rule which assigns to each element of $X$ an element of $Y$.

By unambiguous, I mean that each $x\in X$ has only one element of $Y$ assigned to it. This may seem obvious, but in fact there are some interesting mathematical operations that turn out to be a little ambiguous, for example:

Definition. We say that $t$ is a square root of $x$ if $t^2=x$.

It’s not hard to see that both $-7$ and $7$ are square roots of $49$, so the phrase “the square root of $49$” is actually ambiguous. There are various ways to deal with such ambiguity: for example, we could just content ourselves with the symbol $\pm 7$, representing the two-element set $\{-7,7\}$, but then the square root of a number isn’t a number!. The most straightforward way is just to exclude it by assumption, as I’ve done with the definition of function I gave above.

To functionize the notion of square root, we usually do the following:

Definition. We say that $t$ is the principal square root of $x$, and write $t=\sqrt{x}$, if $t^2=x$ and $t\geq 0$.

That is, if there are the two possible choices for square root, we decide always to take the positive one. This notion of “the square root” is now unambiguous, i.e. a function.

Let’s connect this idea back to our formalism with relations. It’s clear that $t^2=x$ expresses a relationship between $t$ and $x$, so we can hope that our work on relations will help us develop a good definition of function as a kind of relation. For a function from $X$ to $Y$, we will want to consider pairs $(x,y)$, so our relation (let’s be traditional and call it $f$) will be a subset of $X\times Y$.

What would it mean for the rule to be ambiguous? There would be some $x\in X$ with $(x,y_1)\in f$ and $(x,y_2)\in f$, where $y_1$ and $y_2$ are distinct. That is, $f$ being ambiguous means

\begin{equation*}
\exists x\in X,\exists y_1\in Y, \exists y_2\in Y: (x,y_1)\in f \wedge (x,y_2)\in f\wedge y_1\neq y_2
\end{equation*}

Activity. Write a denial of the above logical sentence. This is what it means for $f$ to be unambiguous.

Activity. If your denial only includes $\sim$ and $\vee$, alter it using logical facts so that it contains a $\Rightarrow$.

Overview of Functions

• Ambiguity (this page)
• The basics of functions. Functions as sets and as relations.
• Domains and Rules.
• Operations on functions. How to combine functions.
• Functions and Sets.
• Sets of Functions.
• Functions whose Inverses are Functions.