Set theory/ Cardinal Help
If f : A-B is the given function, the graph f is the subset of AcrossB consisting of all (a,f(a)).
1. Prove that a-(a,f(a)) is a one-to-one map of A onto the graph of f.
2. Let D and E be sets with cardinal numbers d and e. Suppose d is infinite and that e<=d. Prove that d^e is the number of one-to-one maps of E into D.(Hint: of course d^e is an upper bound. To get the reverse inequality, use part(a))
I am OK with question 1.
Here is my trial on Q2
H(E,D) is the set of all functions from E to D. |H(E,D)| = d^e
H'(E,D) is the set of all one-to-one functions from E to D. |H'(E,D)|= n
n <= d^e since H'(E,D) is a subset of H(E,D)
Construct a map F(H(E,D), H'(E,D)) making every function in H(E,D) be a one-to-one function. Then the graph of F is a subset of
H(E,D)crossH'(E,D). Suppose the set of graph F is with cardinal t.
Then, H(E,D) - (H(E,D), H'(E,D)) is a one-to-one map of H(E,D) onto the graph F.
Then we have d^e = t <= d^e * n
I am stuck here. Can we now conclude d^e <= n ? I guess no. But do you guys have any other solutions? Thank you.