Exercise 3.6.5: Let $A$ and $B$ be sets. Show that $A\times B$ and $B\times A$ have equal cardinality by construction an explicit bijection between the two sets. Then use Proposition 3.6.14 (the one about the cardinal arithmetic) to conclude an alternative proof of Lemma 2.3.2 (this lemma proves the commutativity of multiplication).
The bijection is quite easy. But I've no idea why is he asking me to prove a property of multiplication with the cardinality of cartesian products. He defined the natural numbers and its operations with the Peano axioms, no with cardinality, so Tao hasn't really provided a construction of the naturals using only set theory.
What's the point of the exercise? Am I supposed to provide this construction, define the multiplication operation and then prove it or am I missing something?
Best Answer
Following the hint you're given for 3.6.14, I see:
So if you have sets $A,B$ with $|A|=n$ and $|B|=m$, $|A\times B|=n\times m$.
Using exercise 3.6.5, one would also see $|A\times B|=|B\times A|=m\times n$, proving $n\times m = m\times n$ in a different way.