A is a non-empty set. Is there a bijective function such that there exists , with H, and , , is not bijective?
Well, uhm, honestly, I have no idea what to do -_- Should I try to "build" a function?
Thanks in advance.
You are asked whether or not there exist a bijective function that is NOT bijective on a subset. If you believe the answer is yes, the best way to show that would be to create one. If you believe the answer is no, you will have to show why that cannot happen.
O.O Those examples are useless. "You are asked whether or not there exist a bijective function that is NOT bijective on a subset." Better now?
We (I) suppose there is (not) a function g which is not bijective, then:
Let , with , f - injective g - injective
g is not bijective g is (not) surjective
And here I stopped.
it depends on how you define g.
if you are asking if g is a bijection on A, it needn't be.
for example, take A = [-1,1], H = [0,1] and f(x) = x, g(x) = |x|.
if you are asking if g is a bijection on H, only if f(H) = H (if f is not surjective on H, g won't be either).
example: let f(x) = x^3 A = [-1,1], H = [0,1/2], g(x) = f(x).
if you require that f be a bijection on H, then g MUST be one as well.