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Math Help - How to prove a statement of maps and sets?

  1. #1
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    How to prove a statement of maps and sets?

    Hi everyone,
    I need some help on an exercise because i didn't really understand how to prove a statement like this :
    let E and F be two sets and let f be a map from E to F. Let A and A' be two subsets of E. Let B and B' be two subsets of F.

    f(AUA')=( f(A)U f(A'))

    Can you please help me please? I know I have to demonstrate it but I don't know how...

    Thanks a lot
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  2. #2
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    Re: How to prove a statement of maps and sets?

    the usual way to prove 2 sets are equal is to show that each one contains the other.

    so in order to prove that A = B, you show that any a in A is also in B, and that any b in B is also in A.

    so, with this problem, i'll help you get started.

    suppose x is in f(A U A'). this means that x = f(u), for some u that is in A U A'.

    u being in A U A', means that either u is in A, OR u is in A' (or maybe both).

    to show that x is in f(A) U f(A'), you need to show that either x is in f(A) (that is x = f(a) for some a in A), OR x is in f(A') (that is, x = f(a') for some a' in A').

    i recommend you do two cases:

    1) u is in A
    2) u is in A'

    and show that either way...f(u) is in one of f(A) or f(A'). that will be half the proof, then you go "the other way" and start with y in f(A) U f(A').
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  3. #3
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    Re: How to prove a statement of maps and sets?

    Quote Originally Posted by dekl View Post
    Hi everyone,
    I need some help on an exercise because i didn't really understand how to prove a statement like this :
    let E and F be two sets and let f be a map from E to F. Let A and A' be two subsets of E.
    f(AUA')=( f(A)U f(A'))
    I will do half of the above.
    \begin{array}{*{20}c}   {b \in f(A \cup A')}  \\   {\left( {\exists a \in (A \cup A')} \right)\left[ {f(a) = b} \right]}  \\   {\left[ {a \in A \wedge f(a) = b} \right] \vee \left[ {a \in A' \wedge f(a) = b} \right]}  \\   {\left[ {b \in f\left( A \right)} \right] \vee \left[ {b \in f\left( {A'} \right)} \right]}  \\   {b \in \left( {f(A) \cup f(A')} \right)}  \\\end{array}

    Please, you show us the other half.
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  4. #4
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    Re: How to prove a statement of maps and sets?

    Thank you very much, I am working on it.
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    Re: How to prove a statement of maps and sets?

    Quote Originally Posted by Plato View Post
    I will do half of the above.
    \begin{array}{*{20}c}   {b \in f(A \cup A')}  \\   {\left( {\exists a \in (A \cup A')} \right)\left[ {f(a) = b} \right]}  \\   {\left[ {a \in A \wedge f(a) = b} \right] \vee \left[ {a \in A' \wedge f(a) = b} \right]}  \\   {\left[ {b \in f\left( A \right)} \right] \vee \left[ {b \in f\left( {A'} \right)} \right]}  \\   {b \in \left( {f(A) \cup f(A')} \right)}  \\\end{array}

    Please, you show us the other half.
    Sorry but I don't understand why wouldn't it be done like this...
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  6. #6
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    Re: How to prove a statement of maps and sets?

    b is in (f(A) U f(A'))
    then there is an a in A or in A' such that f(a)=b
    then a is in A U B
    so b is in f(A U A')


    is that correct?

    sorry i don't know how to make thi signes...
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  7. #7
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    Re: How to prove a statement of maps and sets?

    Quote Originally Posted by dekl View Post
    b is in (f(A) U f(A'))
    then there is an a in A or in A' such that f(a)=b
    then a is in A U B so b is in f(A U A')
    When proving statement about image is more difficult than is about preimages.
    The reason is the use of the existential operator.
    In this case one would have to say:
    \left( {\exists a \in A} \right)\left[ {f(a) = b} \right] \vee \left( {\exists a' \in A'} \right)\left[ {f(a') = b} \right].
    Now note that \left( {a \in \left( {A \cup A'} \right)} \right) \vee \left( {a' \in \left( {A \cup A'} \right)} \right)
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  8. #8
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    Re: How to prove a statement of maps and sets?

    thank you verr much.

    now can you tell me if I am wrong for the next one?
    f-1(B U B') = (F-1(B) U f-1(B'))
    this means that f(x) B U B' = f(x)B ou f(x) B'
    let take z in B U B' such that f(x)=z
    B U B' means that z is in B or z is in B'
    then we have f(x)B ou f(x) B'
    so f-1(B U B') = (F-1(B) U f-1(B'))
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  9. #9
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    Re: How to prove a statement of maps and sets?

    Quote Originally Posted by dekl View Post
    now can you tell me if I am wrong for the next one?
    f-1(B U B') = (F-1(B) U f-1(B'))
    this means that f(x) B U B' = f(x)B ou f(x) B'
    let take z in B U B' such that f(x)=z
    B U B' means that z is in B or z is in B'
    then we have f(x)B ou f(x) B'
    so f-1(B U B') = (F-1(B) U f-1(B'))
    Yes, your basic idea is correct.

    You can learn LaTeX code.
    [TEX]f^{-1}(B\cup B')=f^{-1}(B)\cup f^{-1}(B')[/TEX] gives
    f^{-1}(B\cup B')=f^{-1}(B)\cup f^{-1}(B').
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