maybe...
dont know for multiplication...
do we look ?
Since so much of cardinal numbers involve creating in/bijections I'm not sure if that's what I should be doing here instead...
Prove that the addition and multiplication is well defined for cardinal numbers.
My answer was... (roughly, more to see if I'm on the right track)
Let A and B be two sets with cardinality and respectively.
Want to show that if card(A) = card(A') and card(B) = card(B') then...
card = card
So we have card = card(A') + card(B') = = card(A) + card(B) = card .
Hence addition is well defined... Seems wrong looking back on it. Do I even need to the bit?
Doing that you assume the addition for cardinal numbers is well-defined.
Moreover, and does not mean ; you must assume to be sure of that (and that's the way cardinal addition is defined).
What you have to prove is: If and then , where for any two sets and .
means there exists a bijection
means there exists a bijection
Question: Can you define a bijection between and ?
For the product, the hypotheses and the question are the same with the cartesian product instead of .
@josipive: The question is about sets in general, you cannot apply such result about vector subspaces of finite dimension.