Let f: A → B be a function. Prove that f is bijective if and only if the inverse relation f^(-1) is a function from B to A. Give a numerical example.

Question

Let f: A → B be a function. Prove that  f is bijective if and only if the inverse relation f^(-1) is a function from B to A. Give a numerical example.

anonymous · Answer

plizzzzzz

b87lar · Answer

(Definition: A function f (from set A to B) is bijective if, for every y in B, there is exactly one x in A such that f(x) = y. )
Proof (=> part): if a function g=f^(-1) exists that maps B to A, then$$f(x)=y$$$$f^{-1}(f(x))=f^{-1}(y)$$$$x=g(y)$$ which implies there is exactly one x for each y with f(x)=y, i.e. f is bijective.
(<= part) if the function f is bijective then by the above definition each element of B has a unique element assigned to it in A. That means there is a bijective function g: B->A such that $$g(y)=x$$Applying composition$$f(g(y))=f(x)$$ and knowing that $$y=f(x)$$it must be that $$g=f^{-1}$$.