Sets, Functions, and Maps

A set is a collection of objects, and a function is a rule that assigns to each element of one set exactly one element of another set. This is the minimal grammar behind vector spaces, operators, wavefunctions, time evolution, coordinate changes, and probability assignments.

The point is not to make quantum mechanics look set-theoretic. The point is to know what a statement such as $A:\mathcal H\to\mathcal H$ actually commits you to: a domain, a codomain, and a rule.

Sets and Elements

If $X$ is a set and $x$ is one of its elements, write

$$x\in X.$$

If every element of $S$ is also an element of $X$, write $S\subseteq X$ and call $S$ a subset of $X$. The empty set is written $\varnothing$.

Common sets include

Symbol	Meaning
$\mathbb N$	natural numbers, with the starting convention stated when needed
$\mathbb Z$	integers
$\mathbb R$	real numbers
$\mathbb C$	complex numbers
$\mathcal H$	Hilbert space

Sets can be built from conditions. For example,

$$S = \{x\in\mathbb R: x^2<1\}$$

is the open interval $(-1,1)$.

Functions and Domains

A function from $X$ to $Y$ is written

$$f:X\to Y.$$

The set $X$ is the domain, and $Y$ is the codomain. For each $x\in X$, the function assigns a single element $f(x)\in Y$.

The codomain is part of the data. The two functions

$$f:\mathbb R\to\mathbb R, \qquad f(x)=x^2,$$

and

$$g:\mathbb R\to[0,\infty), \qquad g(x)=x^2,$$

have the same formula but different codomains. That difference affects whether the map is onto.

Images and Preimages

If $S\subseteq X$, the image of $S$ under $f:X\to Y$ is

$$f(S) = \{f(x)\in Y: x\in S\}.$$

If $T\subseteq Y$, the preimage of $T$ is

$$f^{-1}(T) = \{x\in X: f(x)\in T\}.$$

The notation $f^{-1}(T)$ for a preimage does not require $f$ to have an inverse function. It means all points in the domain that land in $T$.

For example, if $f:\mathbb R\to\mathbb R$ is $f(x)=x^2$, then

$$f^{-1}(\{1\}) = \{-1,1\}.$$

One-to-One, Onto, and Bijective

A function $f:X\to Y$ is:

• injective if different inputs always give different outputs;
• surjective if every element of $Y$ is hit by at least one input;
• bijective if it is both injective and surjective.

Equivalently, $f$ is injective when

$$f(x_1)=f(x_2) \quad\Longrightarrow\quad x_1=x_2,$$

and surjective when for every $y\in Y$ there exists at least one $x\in X$ with

$$f(x)=y.$$

A bijection has a genuine inverse function

$$f^{-1}:Y\to X$$

that reverses the assignment.

Composition and Identity Maps

If

$$f:X\to Y, \qquad g:Y\to Z,$$

then the composition $g\circ f:X\to Z$ is defined by

$$(g\circ f)(x) = g(f(x)).$$

The order matters: $g\circ f$ means apply $f$ first and then $g$.

Each set has an identity map

$$\operatorname{id}_X:X\to X, \qquad \operatorname{id}_X(x)=x.$$

For a bijection $f:X\to Y$, the inverse satisfies

$$f^{-1}\circ f=\operatorname{id}_X, \qquad f\circ f^{-1}=\operatorname{id}_Y.$$

Why Maps Matter in Physics

Quantum mechanics constantly uses maps:

Map	What it does
$\psi:\mathbb R^3\to\mathbb C$	assigns a complex amplitude to each position
$A:\mathcal H\to\mathcal H$	represents a linear operator in a simplified finite-dimensional setting
$U(t):\mathcal H\to\mathcal H$	evolves closed-system states in time
$\rho\mapsto U\rho U^\dagger$	evolves density operators unitarily
$\lvert\psi\rangle\mapsto \lvert\psi\rangle\langle\psi\rvert$	sends a normalized state vector to its rank-one density operator

Some maps are linear, some are nonlinear, some preserve inner products, and some are only defined on a suitable domain. Those extra properties are not automatic; they must be stated.

Worked Example

Let

$$f:\mathbb R\to[0,\infty), \qquad f(x)=x^2.$$

This map is surjective because every $y\in[0,\infty)$ has at least one real square root. It is not injective because

$$f(1)=f(-1)=1$$

while $1\ne -1$.

If instead the domain is restricted to $[0,\infty)$,

$$h:[0,\infty)\to[0,\infty), \qquad h(x)=x^2,$$

then $h$ is bijective and has inverse

$$h^{-1}(y)=\sqrt y.$$

The formula $x^2$ did not change; the domain did. This is why mathematical statements must keep track of domains.

Common Mistakes

• Treating a formula as a complete function without specifying its domain and codomain.
• Confusing the preimage notation $f^{-1}(T)$ with an inverse function.
• Assuming a map is invertible because it is useful.
• Forgetting that composition order matters.
• Treating every physics map as linear.
• Ignoring domains when discussing differential operators or unbounded operators.

Cross-Links

• Mathematical Notation Used in This Volume
• Linear Maps
• Vector Spaces and Dual Spaces
• Inner Products
• Finite-Dimensional Hilbert Spaces
• Tensor Products
• Operators

References

• P. R. Halmos, Naive Set Theory, Springer, 1974.
• S. Axler, Linear Algebra Done Right, 3rd ed., Springer, 2015.
• G. Strang, Linear Algebra and Its Applications, 4th ed., Brooks/Cole, 2006.
• B. C. Hall, Quantum Theory for Mathematicians, Springer, 2013.

Exercises

1. Let $f:\mathbb R\to\mathbb R$ be $f(x)=x^2$. Is $f$ injective? Is it surjective?

Solution

It is not injective because $f(1)=f(-1)$ while $1\ne -1$. It is not surjective onto $\mathbb R$ because no real input maps to a negative real number.

2. For the same function $f:\mathbb R\to\mathbb R$, compute $f^{-1}(\{4\})$.

Solution

The preimage is

$$f^{-1}(\{4\}) = \{-2,2\}.$$

This is a preimage of a set, not an inverse function evaluated at $4$.

3. Let $f:X\to Y$ and $g:Y\to Z$. Which function is applied first in $g\circ f$?

Solution

The function $f$ is applied first:

$$(g\circ f)(x)=g(f(x)).$$