## Properties of Inverse Function

Properties of inverse function are presented with proofs here.
Below f is a function from a set A to a set B.

Property 1:   If f is a bijection, then its inverse f -1 is an injection.

Proof of Property 1:
Suppose that f -1(y1) = f -1(y2) for some y1 and y2 in B.
Then since f is a surjection, there are elements x1 and x2 in A such that y1 = f(x1) and y2 = f(x2).
Then since f -1(y1) = f -1(y2) by the assumption,  f -1(f(x1)) = f -1(f(x2)) holds.
Also by the definition of inverse function, f -1(f(x1)) = x1, and f -1(f(x2)) = x2.
Hence x1 = x2.
Then since f is a function, f(x1) = f(x2), that is y1 = y2.
Thus we have shown that if f -1(y1) = f -1(y2), then y1 = y2.
Hence f -1 is an injection.

QED

Property 2:   If f is a bijection, then its inverse f -1 is a surjection.

Proof of Property 2:
Since f is a function from A to B, for any x in A there is an element y in B such that y= f(x).
Then for that y, f -1(y) = f -1(f(x)) = x, since f -1 is the inverse of f.
Hence for any x in A there is an element y in B such that f -1(y) = x.
Hence f -1 is a surjection.
QED

Property 3:   If f is a bijection, f(f -1(y)) = y for any y in B.

Proof of Property 3:
Since f is a surjection from A to B, for any y in B there is an element x in A such that y= f(x).
Since by the definition of f -1,  f -1(f(x)) = x holds, and since f -1(f(x)) = f -1(y),  f -1(y) = x holds.
Hence f(f -1(y)) = f(x) = y.
Hence f(f -1(y)) = y.
QED

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