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Time to get off your butt and get some exercise!
Jared Summers wrote a nice answer to this one, but there are some natural followups:
Puzzle. If \(A\) and \(B\) are possibly infinite sets, and there's an injection \(f: A \to B\) and an injection \(g: B \to A\), is there a bijection between \(A\) and \(B\)?
Puzzle. If \(A\) and \(B\) are possibly infinite sets, and there's a surjection \(f: A \to B\) and an surjection \(g: B \to A\), is there a bijection between \(A\) and \(B\)?
Vladislav Papayan answered this one, but again there's a subsidiary puzzle:
Puzzle. Given two surjections \(f: A \to P\), \(g: A \to Q\), when do they determine the same partition of \(A\)?
I've also created a bunch of numbered puzzles, and we can work through those. The first three are largely obsolete, luckily, because Fong and Spivak fixed their definition of "poset". Borrowing some text from Matthew Doty:
Puzzle 1. What is a "poset" according to the original version of Chapter 1 of Fong and Spivak's book?
They had defined a poset as a set with a preorder, but now they don't.
Puzzle 2. How does their definition differ from the usual definition found in, e.g., Wikipedia or the nLab?
Ordinarily, a poset is a set with a partial order rather than a preorder.
A partial order is a preorder obeying the antisymmetry axiom: if \(x \le y \) and \( y \le x \) then \( x = y \).
Puzzle 3. What do mathematicians usually call the thing that Fong and Spivak called a poset?
Puzzle 4. List some interesting and important examples of posets that haven't already been listed in other comments in this thread.
Daniel Cellucci has collected many - but probably not all! - our answers here:
To see all the exercises in Chapter 1, go here.