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 $\displaystyle \kappa$ and $\displaystyle \lambda$ respectively.

Want to show that if card(A) = card(A') and card(B) = card(B') then...

card$\displaystyle (A \cup B)$ = card$\displaystyle (A' \cup B')$

So we have card$\displaystyle (A' \cup B')$ = card(A') + card(B') = $\displaystyle \kappa + \lambda$ = card(A) + card(B) = card$\displaystyle (A \cup B)$.

Hence addition is well defined... Seems wrong looking back on it. Do I even need to the $\displaystyle \kappa, \lambda$ bit?