Images and Inverse Images

Functions act on individual elements — can we lift them to act on whole sets? Yes, in two natural ways. Push a subset of the domain forward: collect every output you reach. That is the image. Pull a subset of the codomain back: collect every input that lands there. That is the inverse image. The two operations look symmetric but are not: preimages behave perfectly under all set operations, while images preserve unions but only partially preserve intersections. This asymmetry, once you see it, explains a small but persistent collection of "obvious" facts that turn out to be false.

Image of a set

For $f: A \to B$ and $S \subseteq A$, the image of $S$ is

It is the set of all outputs you get by feeding elements of $S$ to $f$.

Inverse image (preimage)

For $T \subseteq B$, the inverse image (or preimage) of $T$ is

Crucial: $f^{-1}(T)$ is defined for every function, no bijection required — this $f^{-1}$ is set-valued, not the inverse function of §5.3.

Preimage commutes with everything

Preimages preserve union, intersection, and complement perfectly:

Image commutes only with union

Images preserve unions:

But for intersections, only one direction holds:

and equality can fail. Example: $f(x) = x^2$ with $S_1 = {-1}, S_2 = {1}$. Then $S_1 \cap S_2 = \emptyset$ so $f(S_1 \cap S_2) = \emptyset$, but $f(S_1) \cap f(S_2) = {1} \cap {1} = {1}$. Equality of $f(S_1 \cap S_2)$ and $f(S_1) \cap f(S_2)$ holds for every $S_1, S_2$ iff $f$ is injective.

(The widget above shows the arrows of a function. To visualise an image, highlight the source elements in $S$ and trace their arrow heads; the set of distinct heads IS $f(S)$. For a preimage of $T$, highlight target elements in $T$ and trace arrows backwards.)

Pause and think: If $f$ is injective, can you prove $f(S_1 \cap S_2) = f(S_1) \cap f(S_2)$? Where does injectivity get used? (Hint: start with $y \in f(S_1) \cap f(S_2)$, write $y = f(a_1) = f(a_2)$, apply injectivity to conclude $a_1 = a_2 \in S_1 \cap S_2$.)

Try it

• For $f(x) = x^2$ on $\mathbb{R}$, compute $f({-2, -1, 0, 1, 3})$. Predict the size before listing.
• For $f(x) = 2x + 1$ on $\mathbb{R}$, compute $f^{-1}({3, 7, 11})$.
• For $f(x) = |x|$ on $\mathbb{R}$, exhibit subsets $S_1, S_2$ where $f(S_1 \cap S_2) \subsetneq f(S_1) \cap f(S_2)$.
• Prove $f^{-1}(T_1 \cap T_2) = f^{-1}(T_1) \cap f^{-1}(T_2)$ by element-chasing.
• True or false: $f(f^{-1}(T)) = T$ for every $T \subseteq B$. (Answer: only one inclusion holds in general — which one?)

A trap to watch for

"Image preserves intersection" is the most common false friend in this section. Students assume that $f(S_1 \cap S_2) = f(S_1) \cap f(S_2)$ because preimages do this for free — but images only get a one-sided inclusion. Whenever you write that equality, you are silently using injectivity; if $f$ is not injective, you need a counterexample-ready mindset. The diagnostic question: "could two different inputs collide on the same output?"

What you now know

You can push subsets through a function in both directions, prove the union and intersection laws, and pinpoint exactly where the image-vs-preimage asymmetry breaks (intersection for images, fixed by injectivity). With images, preimages, functions, relations, equivalences, and orderings all in your toolkit, you have the full set-theoretic vocabulary that Chapter 6 will deploy to define the natural numbers from scratch and prove theorems by induction.

References

• Velleman, D. J. (2019). How to Prove It (3rd ed.). Cambridge University Press, §5.5.
• Halmos, P. R. (1960). Naive Set Theory. Springer, §8-9.
• Hammack, R. (2018). Book of Proof, ch. 12.
• Bloch, E. D. (2011). Proofs and Fundamentals (2nd ed.). Springer, ch. 4.
• Goodfellow, I., Shlens, J., Szegedy, C. (2015). "Explaining and harnessing adversarial examples." ICLR 2015 (preimages and adversarial inputs).