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Anonymous
Anonymous asked in 科學及數學數學 · 10 years ago

Problem about set theory

Set A and B are defined as follows:

A={x: 1<=x<9 } and B={y: y<=5}.

a) Are Sets A and B convex? Are they closed? Are the bounded?

b) Is the set A∩B convex, closed and/or bounded?

Update:

Could you please explain more?

Update 2:

This one:

unbounded( y can approach to -∞)

1 Answer

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  • 10 years ago
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    (a)

    A is convex {x1, x2 in [1, 9), a+b=1, a, b>=0, then ax1+bx2 in [1, 9) }

    not closed(not contain the limit point x=9),

    bounded ( lower bound=1 , upper bound=9)

    B is convex, closed(contain any limit points), unbounded

    (b)

    A∩B is convex (intersection of convex sets A and B)

    not closed [not contain the limit point (9, 5) ]

    unbounded( y can approach to -∞)

    2010-10-09 13:38:54 補充:

    Which one?

    2010-10-10 19:44:39 補充:

    A∩B={(x,y)| 1 <= x < 0, y <= 5}

    If A∩B is bounded then there exists M>0 such that x^2+y^2 < M for all (x,y) in A∩B.

    i.e. A∩B is enclosed by a circle, it is impossible, since y can approach to -∞.

    Thus, A∩B is unbounded.

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