## 502 – Cancellation laws

Two homework problems. The first one is easier, so you can consider the second one to be extra credit. A proof of these results can be found in different places, for example, the paper Division by three, by Conway and Doyle. (Please don’t look at the paper while working on the homework, of course.) Unfortunately, the paper could use a serious trimming and editing, so I cannot really recommend it, but the proof is carefully written there.

1. Without using the axiom of choice, show that if $A$ and $B$ are sets, and $|A\times 2|=|B\times 2|,$ then $|A|=|B|.$
2. Same as 1., but now with $3$ instead of $2.$