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Show that if f: A->B, and A(1), A(2) are both subsets of A, then

  1. Sep 15, 2010 #1
    Show that if f: A-->B, and A(1), A(2) are both subsets of A, then

    Show that if f: A-->B, and A(1), A(2) are both subsets of A, then
    f(A1 ∩ A2) C(is the subset of) f(A1) ∩ f(A2).

    Give an example of a situation where the inclusion is strict.
     
  2. jcsd
  3. Sep 15, 2010 #2
    Re: Show that if f: A-->B, and A(1), A(2) are both subsets of A, then

    For every element x in f(A1 ∩ A2), there must be an element y in A1 ∩ A2 mapping through f to x... and continue from there...

    You need to show some semblance of work before we can really help you on a problem.
     
  4. Sep 15, 2010 #3
    Re: Show that if f: A-->B, and A(1), A(2) are both subsets of A, then

    I really don't understand the whole proof
     
  5. Sep 15, 2010 #4

    jgens

    User Avatar
    Gold Member

    Re: Show that if f: A-->B, and A(1), A(2) are both subsets of A, then

    You haven't posted a proof, so we can't help you understand the "proof". If you mean that you don't understand the problem, what about it is tripping you up?
     
  6. Sep 15, 2010 #5
    Re: Show that if f: A-->B, and A(1), A(2) are both subsets of A, then

    Sorry i meant to say i don't quite understand the problem. However i have gotten this much of understanding which i do know if it is right.

    Let x be an element of f(A1 ∩ A2) and by definition of the f(A1 ∩ A2), there is a y element in ( A1 ∩ A2) so that f(y)=x.
    Since y is an element in (A1 ∩ A2), y∈A1x∈A2. Since y,f(y)∈ f(A1).
    This follows alongside y,f(y)f(A2)
    and
    Since f(y)=x∈f(A1) and f(y)=x∈f(A2),x= f(A1)(f(A2)

    I don't know how to complete it
    I would appreciate you help thank u
     
  7. Sep 15, 2010 #6
    Re: Show that if f: A-->B, and A(1), A(2) are both subsets of A, then

    Your reasoning seems to be right. All you need to show is that any element x in f(A1 ∩ A2) is also in f(A1) ∩ f(A2). Do you understand why your argument does this?
     
  8. Sep 15, 2010 #7
    Re: Show that if f: A-->B, and A(1), A(2) are both subsets of A, then

    How about f(x)=|x|, A11=R-,A2=R+. Any noninjective function really.
     
  9. Sep 15, 2010 #8
    Re: Show that if f: A-->B, and A(1), A(2) are both subsets of A, then

    Though am i correct about the PROOF( for f(A1 ∩ A2) ⊆ f(A1) ∩ f(A2) )
    that i have shown below???

    Let x∈A1∩A2. Then x∈A1 or x∈A2, in which case f(x)∈f(A1) or f(A2) respectively; in any case, f(x)∈f(A1)∩f(A2), and so f(A1∩A2)⊆f(A1)∩f(A2).

    On the other hand, let y∈f(A1)∩f(A2). Then y∈f(A1) or y∈f(A2), in which case there is some x∈A1 or A2 respectively with f(x)=y. Hence y∈f(A1∩A2), whence f(A1)∩f(A2)⊆f(A1∩

    Therefore f(A1∩A2)⊆((A1)∩f(A2).
     
    Last edited: Sep 15, 2010
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