o f = A and f o
= B, where f is a bijective function and f^-1 is its inverse function.
Let f: A → B be a bijective function, and let
: B → A be its inverse function.
We need to show that
o f = A and f o
= B.
To prove
o f = A, we need to show that (
o f)(x) = x for all x in A.
Let x be an arbitrary element in A. Since f is a bijective function, there exists a unique y in B such that f(x) = y. Therefore, we can write:
(
o f)(x) =
(f(x))
=
(y)
Since
is the inverse function of f, we know that
(y) = x. Therefore, we can write:
(
o f)(x) = x
This holds for all x in A, so we have shown that
o f = A.
To prove f o
= B, we need to show that (f o f^-1)(y) = y for all y in B.
Let y be an arbitrary element in B. Since f is a bijective function, there exists a unique x in A such that f(x) = y. Therefore, we can write:
(f o
)(y) = f(
(y))
= x
Since
is the inverse function of f, we know that
(y) = x. Therefore, we can write:
(f o
)(y) = y
This holds for all y in B, so we have shown that f o
= B.
Therefore, we have proven that
o f = A and f o
= B, where f is a bijective function and
is its inverse function.
Complete question: Let