Find a relationship between ff 1fA and fA a Find an example

Find a relationship between f(f ^-1(f(A))) and f(A).

a. Find an example where your relationship works.

b. Prove it in general.

Let , a belong to f(A)

There there is some x in f-1(f(A)) so that

f(x)=a

Hence, a belong to f(f-1(f(A)))

So, f(A) is a subset of f(f-1(f(A)))

Let, a belong to f(f-1(f(A)))

So,

f(y)=a for some ,y(there can be more than one element here so that f(z)=a) in f^{-1}(f(A))

y in f^{-1}(f(A))

Hence, f(y) =a is in f(A)

Hence, f(A)=f(f-1(f(A)))

This is a proof for part b.

Let, A={1,2,3}

f=id_A ie identity map on A

f^{-1}=f=id_A

f(A)=A

f(f-1((f(A)))=A