# Determining if factorization function is onto

Here's a question from my math textbook:

Let $D$ be the set of all finite subsets of positive integers, and define $T : \mathbb{Z}^+ \to D$ by the rule: For all integers $n$, $$T (n) = \text{the set of all of the positive divisors of }n.$$ a. Is $T$ one-to-one? Prove or give a counterexample.
b. Is $T$ onto? Prove or give a counterexample.

The answer in the back claims that $T$ is neither one-to-one nor onto. Onto is obvious, but how could $T$ not be one-to-one? No two numbers have the same factorization. Is this a book error?

## 2 Answers

You are correct: $T$ is one-to-one. In fact, since $\max T(n)=n$, we see that $\max$ is a left inverse of $T$.

$T$ is not onto because sets of divisors form a lattice under divisibility and not all finite subsets of $\mathbb N$ are lattices.

• Or one could just observe that any subset not containing $1$ is not in the image. – William Stagner Sep 30 '15 at 1:13