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Homework Help: Are there any kinds of functions which satisfy f(A1 ∩ A2) =f(A1) ∩ f(A2)? Prove your

  1. Sep 15, 2010 #1
    Are there any kinds of functions which satisfy f(A1 ∩ A2) =f(A1) ∩ f(A2)? Prove your claim.?
     
  2. jcsd
  3. Sep 15, 2010 #2
    Re: Are there any kinds of functions which satisfy f(A1 ∩ A2) =f(A1) ∩ f(A2)? Prove y

    The identity function f(x) = x
     
  4. Sep 15, 2010 #3
    Re: Are there any kinds of functions which satisfy f(A1 ∩ A2) =f(A1) ∩ f(A2)? Prove y

    The identity function?
     
  5. Sep 15, 2010 #4
    Re: Are there any kinds of functions which satisfy f(A1 ∩ A2) =f(A1) ∩ f(A2)? Prove y

    This is true for every f:

    [tex]f(A\cap B)\supset f(A)\cap f(B)[/tex]

    This is true for every injective f:

    [tex]f(A\cap B)\subset f(A)\cap f(B)[/tex]

    So the answer to your question is: yes, every injective function.
     
  6. Sep 15, 2010 #5
    Re: Are there any kinds of functions which satisfy f(A1 ∩ A2) =f(A1) ∩ f(A2)? Prove y

    Yes, namely: all one to one functions.

    Now try proving this and show us your work.
     
  7. Sep 15, 2010 #6

    Mark44

    Staff: Mentor

    Re: Are there any kinds of functions which satisfy f(A1 ∩ A2) =f(A1) ∩ f(A2)? Prove y

    Please do not double-post your questions.
     
  8. Sep 15, 2010 #7
    Re: Are there any kinds of functions which satisfy f(A1 ∩ A2) =f(A1) ∩ f(A2)? Prove y

    In order to show that it is a one-to-one function( Injective) i have go the following steps but i don't know where to go with it after it is confusing....


    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)
     
  9. Sep 15, 2010 #8
    Re: Are there any kinds of functions which satisfy f(A1 ∩ A2) =f(A1) ∩ f(A2)? Prove y

    In this case, you want to show the other direction. That, if x is in f(A1) ∩ f(A2) and f is injective, then x is also in f(A1 ∩ A2).
     
  10. Sep 15, 2010 #9
    Re: Are there any kinds of functions which satisfy f(A1 ∩ A2) =f(A1) ∩ f(A2)? Prove y

    So can you tell me whether i am correct now??

    So, let y∈f(A1)∩f(A2); then y∈f(A1) and y∈f(A2). Then there is an x1∈A1 and an x2∈A2 with f(x1)=f(x2)=y. But since f is one-to-one, x1=x2, and so y∈f(A1∩A2), completing the proof.
     
  11. Sep 15, 2010 #10
    Re: Are there any kinds of functions which satisfy f(A1 ∩ A2) =f(A1) ∩ f(A2)? Prove y

    That's correct.
     
  12. Sep 15, 2010 #11
    Re: Are there any kinds of functions which satisfy f(A1 ∩ A2) =f(A1) ∩ f(A2)? Prove y

    Thank You:)
     
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