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Math Help - Examples for properties of Sets

  1. #1
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    Smile Examples for properties of Sets

    I have three problems that I can't come up with examples for.

    1.) Give an example of two sets, E1 and E2, such that closure(E1 intersect E2) does not equal closure(E1) intersect closure(E2)

    2.) Give an example of two sets, E1 and E2, such that int(E1 U E2) does not equal int(E1) U int(E2)

    3.) Give an example of a sequence of closed sets F1, F2, F3,... whose union is neither open nor closed.

    I was told that any open set like (1,3) and a closed set [2,4] would work for number 1, but then closure(E1 intersect E2)=[2,3] and closure(E1) intersect closure(E2) = [2,3] so I do not know what to do.

    I can't come up with anything else for any of them, so any help would be greatly appreciated.
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  2. #2
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    Quote Originally Posted by xoyankeegirlx3 View Post
    1.) Give an example of two sets, E1 and E2, such that closure(E1 intersect E2) does not equal closure(E1) intersect closure(E2)

    2.) Give an example of two sets, E1 and E2, such that int(E1 U E2) does not equal int(E1) U int(E2)

    3.) Give an example of a sequence of closed sets F1, F2, F3,... whose union is neither open nor closed.
    Let E_1=(0,1]~\&~E_2=(1,2).
    Do they work for both 1 & 2?
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  3. #3
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    Quote Originally Posted by xoyankeegirlx3 View Post
    3.) Give an example of a sequence of closed sets F1, F2, F3,... whose union is neither open nor closed.
    Consider sets of the form [a,b-1/n]
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  4. #4
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    1) try (0,2) and (2,4). Their intersection is { } so closure is empty set right? Then closure of (0,2) is [0,2] intersect closure (2,4) is [2,4] and you get an intersection of 2.

    2)try int([0,1] u [1,2]) = int [0,2] = (0,2) and int[0,1] u int[1,2] = (0,1) u (1,2)
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  5. #5
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    Quote Originally Posted by Plato View Post
    Let E_1=(0,1]~\&~E_2=(1,2).
    Do they work for both 1 & 2?
    I don't think..I think both answers for both 1 and 2 both come out to [1,2]
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  6. #6
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    Quote Originally Posted by Focus View Post
    Consider sets of the form [a,b-1/n]
    Awesome!! Thank you, those intervals worked perfectly!
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  7. #7
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    Quote Originally Posted by mathtasteslikepi View Post
    1) try (0,2) and (2,4). Their intersection is { } so closure is empty set right? Then closure of (0,2) is [0,2] intersect closure (2,4) is [2,4] and you get an intersection of 2.

    2)try int([0,1] u [1,2]) = int [0,2] = (0,2) and int[0,1] u int[1,2] = (0,1) u (1,2)
    Thank you! Exactly what I was looking for!
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  8. #8
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by xoyankeegirlx3 View Post
    I don't think..I think both answers for both 1 and 2 both come out to [1,2]
    I do believe Plato is correct...
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