Question

Let f : A → B, IA and IB be identify functions on A and B respectively. Then prove that foIA= f = IBof.

Answer 1

Given f : A → B be a function

Also IA : A → A and IB : B → B be identity functions

IA : A → A, f : A → B ⇒ foIA : A → B

∴ foIA and f are defined on same domain A.

Let a ∈ A

Then (foIA) (a) = f[IA(a)]

= f(a)

∴ foIA = f → (1)

f : A → B, IB : B → B ⇒ IBoF : A → B

∴ IBof and f are defined on the same domain A.

Let a ∈ A

Then (IBof) (a) = IB[f(a)] = f(a)

∴ IBof = f → (2)

From (1) and (2)

foIA = f = IBof