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Sets and functions proofs needed

  1. Sep 28, 2012 #1
    Hello there,

    I am extremely new to mathematical analysis and do not have an idea how to prove the following questions. Could you please give me a hand and show me a way?

    Let At , t ∈ T, be a family of sets, and let X be a set. Prove the identities


    http://desmond.imageshack.us/Himg62/scaled.php?server=62&filename=98278507.png&res=landing [Broken]


    Let A and B are sets, and let f : A → B be a function defined in A with values in B.

    Is it true that f (A \ B) ⊂ f (A) \ f (B)?
    Is it true that f (A \ B) ⊃ f (A) \ f (B)
     
    Last edited by a moderator: May 6, 2017
  2. jcsd
  3. Sep 28, 2012 #2
    Do you know how to prove that two sets are equal? If A and B are sets and I want to prove that A = B, I have to show two things: That A ⊂ B and B ⊂ A.

    And to show each of those two things, you have to show that

    a) If x is an element of A, then x is an element of B.

    and

    b) If x is an element of B, then x is an element of A.

    Does any of this sound familiar? Can you apply it to your problems?
     
    Last edited by a moderator: May 6, 2017
  4. Sep 28, 2012 #3
    Thank you so much for your instant reply. I think I understood your point. For example for the first one:

    left to right:
    if m∈ X∖⋃At => m ∈ X and m∉⋃At. so m∉At for all t∈At.
    if m∈X and m∉At for all t∈At, then X\At={m} for al t∈T
    then, m∈⋂(X∖At)


    if m∈⋂(X∖At) => m ∈ X\At for all t∈T. then, m∈X, m∉At for all t∈At.
    if m∉At for all t∈At => m∉⋃At.
    if m∈X and m∉⋃At => m∈X∖⋃At

    Is my notation true?
     
  5. Sep 28, 2012 #4

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Excellent!
    No, you can't say "X\At= {m}", you don't know if there aren't other elements in X\At. You can, of course, say "m∈ X\At for all t" and that's all you need.

    Yes, this is exactly right.

     
  6. Sep 28, 2012 #5
    and for the second one:

    left to right:

    if m ∈ X\⋂At => m∈X and m∉⋂At => m∈X and m∉At for all t∈T => m∈X\At for for all t∈T. Therefore m∈⋃(X\At)

    right to left:

    if m∈⋃(X\At) => m∈X and (∃t∈T that m∈At or ∄t∈T that m∈At)
    if not for all t∈T, m∈At, then m∉⋂At. therefore m∈X\⋂At


    I feel like my notations are not very good. How can I do these solutions better?
     
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