1.4.8: Equinumerosity
We have an intuitive notion of “size” of sets, which works fine for finite sets. But what about infinite sets? If we want to come up with a formal way of comparing the sizes of two sets of any size, it is a good idea to start by defining when sets are the same size. Here is Frege:
If a waiter wants to be sure that he has laid exactly as many knives as plates on the table, he does not need to count either of them, if he simply lays a knife to the right of each plate, so that every knife on the table lies to the right of some plate. The plates and knives are thus uniquely correlated to each other, and indeed through that same spatial relationship. ( Frege, 1884 , §70)
The insight of this passage can be brought out through a formal definition:
\(A\) is equinumerous with \(B\) , written \(\cardeq{A}{B}\) , iff there is a bijection \(f \colon A \to B\) .
Equinumerosity is an equivalence relation.
Proof. We must show that equinumerosity is reflexive, symmetric, and transitive. Let \(A, B\) , and \(C\) be sets.
Reflexivity. The identity map \(\Id{A} \colon A \to A\) , where \(\Id{A} (x) = x\) for all \(x \in A\) , is a bijection. So \(\cardeq{A}{A}\) .
Symmetry. Suppose \(\cardeq{A}{B}\) , i.e., there is a bijection \(f\colon A \to B\) . Since \(f\) is bijective, its inverse \(f^{-1}\) exists and is also bijective. Hence, \(f^{-1}\colon B \to A\) is a bijection, so \(\cardeq{B}{A}\) .
Transitivity. Suppose that \(\cardeq{A}{B}\) and \(\cardeq{B}{C}\) , i.e., there are bijections \(f\colon A \to B\) and \(g\colon B \to C\) . Then the composition \(\comp{f}{g}\colon A \to C\) is bijective, so that \(\cardeq{A}{C}\) . ◻
If \(\cardeq{A}{B}\) , then \(A\) is countable if and only if \(B\) is.
Proof. Suppose \(\cardeq{A}{B}\) , so there is some bijection \(f \colon A \to B\) , and suppose that \(A\) is countable. Then either \(A = \emptyset\) or there is a surjective function \(g\colon \PosInt \to A\) . If \(A = \emptyset\) , then \(B = \emptyset\) also (otherwise there would be an element \(y \in B\) but no \(x \in A\) with \(g(x) = y\) ). If, on the other hand, \(g\colon \PosInt \to A\) is surjective , then \(\comp{f}{g} \colon \PosInt \to B\) is surjective . To see this, let \(y \in B\) . Since \(g\) is surjective , there is an \(x \in A\) such that \(g(x) = y\) . Since \(f\) is surjective , there is an \(n \in \PosInt\) such that \(f(n) = x\) . Hence, \[(\comp{f}{g})(n) = g(f(n)) = g(x) = y\nonumber\] and thus \(\comp{f}{g}\) is surjective . We have that \(\comp{f}{g}\) is an enumeration of \(B\) , and so \(B\) is countable .
If \(B\) is countable, we obtain that \(A\) is countable by repeating the argument with the bijection \(f^{-1}\colon B \to A\) instead of \(f\) . ◻
Show that if \(\cardeq{A}{C}\) and \(\cardeq{B}{D}\) , and \(A \cap B = C \cap D = \emptyset\) , then \(\cardeq{A \cup B}{C \cup D}\) .
Show that if \(A\) is infinite and countable, then \(\cardeq{A}{\Nat}\) .