Here's the problem set:
Let A = {1,2,3,4,5,6}. In each of the following, give an example of a function f : A → A with the indicated properties, or explain why no such function exists.
(a) f is bijective, but is not the identity function f(x) = x.
(b) f is neither one-to-one nor onto.
(c) f is one-to-one, but not onto.
(d) f is onto, but not one-to-one.
I'm having trouble with this one because having a finite domain and codomain seem very "rigid" to me in the sense that f : A → A where A is finite is inherently bijective, so I'm not sure how to deviate from that. Here's where I'm at so far:
a) Doesn't exist because for f to stay bijective, it has to be the identity function.
b) Doesn't exist because I can't write f(x) = 10 because 10 isn't part of A.
c) Doesn't exist because f : A → A is inherently bijective.
d) Doesn't exist because f : A → A is inherently bijective.
