# Thread: Inverse function of intersection = intersection of inverse functions

1. ## Inverse function of intersection = intersection of inverse functions

From Rosen sixth edition, page 148, section 2.3, 40.b.
Let f be a function from A to B. Let S and T be subsets of B. Show that

f inverse(S intesect T) = f inverse (S) intersecting f inverse (T)

I think I'm starting to get the hang of this but please advise on the correctness!

f:A->B S subset of B, T subset of B

Let a be an arbitrary element of f inverse ( S intersecting T) then f(a) is an element of S intersecting T

so f(a) element of S AND f(a) element of T

then a element of f inverse (S) AND a element of f inverse (T)

so a element of (f inverse (S) AND f inverse (T))

finally a element of f inverse (S) intersecting f inverse (T)

Since a is an arbitrary element of f inverse (S intersecting T) and
also of (F inverse (S) intersecting f inverse (T)) this completes the proof.

Please tell me if and where I've gone wrong.

Thanks

2. Originally Posted by oldguynewstudent
From Rosen sixth edition, page 148, section 2.3, 40.b.
Let f be a function from A to B. Let S and T be subsets of B. Show that

f inverse(S intesect T) = f inverse (S) intersecting f inverse (T)

I think I'm starting to get the hang of this but please advise on the correctness!

f:A->B S subset of B, T subset of B

Let a be an arbitrary element of f inverse ( S intersecting T) then f(a) is an element of S intersecting T

so f(a) element of S AND f(a) element of T

then a element of f inverse (S) AND a element of f inverse (T)

so a element of (f inverse (S) AND f inverse (T))

finally a element of f inverse (S) intersecting f inverse (T)

Since a is an arbitrary element of f inverse (S intersecting T) and
also of (F inverse (S) intersecting f inverse (T)) this completes the proof.

Please tell me if and where I've gone wrong.

Thanks
You have proved if a belong to f inverse(S intesect T), it belongs to f inverse (S) intersecting f inverse (T)
Thus f inverse(S intesect T) is a subset of f inverse (S) intersecting f inverse (T)

What about the other way round. That still needs to be established for equality - correct?

3. Originally Posted by aman_cc
You have proved if a belong to f inverse(S intesect T), it belongs to f inverse (S) intersecting f inverse (T)
Thus f inverse(S intesect T) is a subset of f inverse (S) intersecting f inverse (T)

What about the other way round. That still needs to be established for equality - correct?
Yes, I need to prove the other way for complete proof.

Thanks, I have a test on Thursday and knowing this is correct will really help!