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Math Help - General and subsets

  1. #1
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    General and subsets

    Have to determine whether each one is true or false.

    For #6. I used a similar proof for F(AUB). Had this:
    suppose y∈(A⋂B)={F(x)|x∈A⋂B}
    Then y=F(x) for some x∈A⋂B
    y=F(x) where x∈A or x∈B
    But x∈A---> y=F(x)∈F(A)
    and x∈B--> y=F(x)∈F(B)
    y∈F(A) and y∈F(B)
    y∈F(A)⋂F(B)

    is this correct or is this different way for ⋂?
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  2. #2
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    Quote Originally Posted by hellfire127 View Post
    Have to determine whether each one is true or false.

    For #6. I used a similar proof for F(AUB). Had this:
    suppose y∈(A⋂B)={F(x)|x∈A⋂B}
    Then y=F(x) for some x∈A⋂B
    y=F(x) where x∈A or x∈B
    But x∈A---> y=F(x)∈F(A)
    and x∈B--> y=F(x)∈F(B)
    y∈F(A) and y∈F(B)
    y∈F(A)⋂F(B)

    is this correct or is this different way for ⋂?
    Your proof is incomplete for number 6. You have proven that F\left(A\cap B\right)\subseteq A\cap B. But, the reverse inclusion is not true. Consider a non-injective function (such as -1,1)\to(-1,1):x\mapsto x^2" alt="f-1,1)\to(-1,1):x\mapsto x^2" />).

    For the second one, are you having trouble? If so, where at?
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  3. #3
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    yea just found out that number 6 is false using A as {1,} and B as {1,3} for example.

    yes, i am having trouble understading the inverse for C and D on number 7.
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  4. #4
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    Quote Originally Posted by hellfire127 View Post
    yea just found out that number 6 is false using A as {1,} and B as {1,3} for example.

    yes, i am having trouble understading the inverse for C and D on number 7.
    What do you mean the 'inverse'?
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  5. #5
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    Quote Originally Posted by Drexel28 View Post
    What do you mean the 'inverse'?
    the power to -1
    F^-1
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  6. #6
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by hellfire127 View Post
    the power to -1
    F^-1
    It's not a literal inverse, it's the inverse image. Namely, F^{-1}\left(A\right)=\left\{x:F(x)\in A\right\}
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  7. #7
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    yea meant inverse image. i really need to go back and read the section over on this as I don't even know how to start most of these.
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  8. #8
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by hellfire127 View Post
    yea meant inverse image. i really need to go back and read the section over on this as I don't even know how to start most of these.
    Well, what if I get you started?

    If x\in F^{-1}(U\cap V\right) then F(x)\in U\cap V and so F(x)\in U\text{ and }F(x)\in V and so x\in F^{-1}(U)\text{ and }x\in F^{-1}(V) and so x\in F^{-1}(U)\cap F^{-1}(V).
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  9. #9
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    Quote Originally Posted by Drexel28 View Post
    Well, what if I get you started?

    If x\in F^{-1}(U\cap V\right) then F(x)\in U\cap V and so F(x)\in U\text{ and }F(x)\in V and so x\in F^{-1}(U)\text{ and }x\in F^{-1}(V) and so x\in F^{-1}(U)\cap F^{-1}(V).
    so whenever there is an inverse image for x\in F^{-1}(A\cap B\right) then F(x)\in A\cap B ? like the A and B can be any sets correct?
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  10. #10
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by hellfire127 View Post
    so whenever there is an inverse image for x\in F^{-1}(U\cap V\right) then F(x)\in A\cap B ? like the A and B can be any sets correct?
    Yes, because this is the definition.
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  11. #11
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    ok this seems a lot harder compared to mathematical induction earlier.

    my book has a smiliar problem to number 6 but i don't understand its prove. do you have a better explaination for number 6? i just can't seem to see how it works out. i understand your solution to number 7 more.
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  12. #12
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by hellfire127 View Post
    ok this seems a lot harder compared to mathematical induction earlier.

    my book has a smiliar problem to number 6 but i don't understand its prove. do you have a better explaination for number 6? i just can't seem to see how it works out. i understand your solution to number 7 more.
    What do you mean explanation? You proved that the inclusion goes one way, but not the other. What more do you seek?
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  13. #13
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    like something you did smilar to number 7 for number 6.
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  14. #14
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by hellfire127 View Post
    like something you did smilar to number 7 for number 6.
    You already did that yourself.
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  15. #15
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    Quote Originally Posted by Drexel28 View Post
    You already did that yourself.
    ok so 6 is true by what i did on first post? then what did you mean by the reverse inclusion? that F(A)⋂(FB)=F(A⋂B) is false right?
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