I've been given a question on my problem set which I'm a bit stuck with. I'm getting the sense that the first part is easier than the second part, but I can't even seem to get the first part!
a) Let A, X, Y be sets such that . Show that . Therefore show that, for cardinals , if then .
b) Let A, B, X, Y be sets with and . Show that, apart from some exceptional cases, . What are the exceptional cases?
I'd really appreciate any help you can provide!
Here's something to get you started for the second one. Let f be an injection from X to A, and g an injection from Y to B. Now, let h be an arbitrary function from Y to X. The most natural thing to do is let F(h) be the function k from B to A where to compute k(b), you take if possible (this gives an element of Y), then apply h (this gives an element of X), and finally apply f (this gives an element of A). Now think about how you would define k on elements not in the range of g. Going through the argument carefully will probably shed some light on the exceptional cases.