Math 3345 Section 16 Exercise 9
Let A, B, C, D be sets such that A is equinumerous to C and B is equinumerous to D. Show that AxB is equinumerous to CxD.
I believe I have the correct answer for when all A, B, C, D are finite. I think it works but I'm not sure if it's correct for when the sets are infinite.
My answer: I'm using the '||' symbols to denote "the cardinality of"/"the length of", and 'x' to represent Cartesian product (I apologize, I'm new to formatting on the site)
A equinumerous to C, therefore |A| = |C|
B equinumerous to D, therefore |B| = |D|
|AxB| = (|A|)(|B|) = (|C|)(|B|) = (|C|)(|D|) = |CxD|
Therefore, since |AxB| = |CxD|, then AxB is equinumerous to CxD.