Discrete Mathematical Structures by Pearson

Functions 3-17

3.9 SOME IMPORTANT THEOREMS AND PROBLEMS

THEOREM 3.6 If f: A → B is one-one and ONTO, then the inverse mapping of f is unique.

Proof: Let f: A → B be one-one and ONTO.

Let g: B → A and h: B → A be inverse mappings of f.

Let b be any element in B and let g(b) = c and h(b) = d, where c and d ∈ A.

Since g is the inverse mapping of f, then g(b) = c ⇒ f (c) = b

Similarly, since h is inverse mapping of f, so, h(b) = d ⇒ f (d) = b

∴ b = f (c) = f (d) ⇒ c = d since f is one-one.

Hence,

c = d ⇒ g(b) = h(b)

⇒ g = h

Since g is the same function as h and so the inverse mapping of f is unique.

THEOREM 3.7 Let f: A → B and g: B → A be two functions such that gof: A → A is the identity

function on I

A

, then g = f

-1

. If fog: B → B is the identity function ...