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Math Help - The union of two convex sets is not convex

  1. #1
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    The union of two convex sets is not convex

    Can I demonstrate, using Venn Diagrams, that a union of two convex sets is not necessarily convex simply by drawing something like this and then drawing a line from the top of one circle to the top of another? (The line would go outside the circles, indicating the union is not convex.)
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  2. #2
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    Quote Originally Posted by Gemini View Post
    Can I demonstrate, using Venn Diagrams, that a union of two convex sets is not necessarily convex simply by drawing something like this and then drawing a line from the top of one circle to the top of another? (The line would go outside the circles, indicating the union is not convex.)

    Some wouldn't accept a drawing as a proof (I, for one... ), but you can easily build an easy example: take the two tangential circles

    (x-1)^2+y^2=1\,,\,\,(x+1)^2+y^2=1 ,and now prove that the line joining the points (1,1\slash 2)\,,\,\,(-1,1\slash 2) is not completely contained in the union of these two circles.

    Tonio
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  3. #3
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    Thanks, but if my example is correct, I'll just use it. If you don't want drawings, you better tell it
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  4. #4
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    Quote Originally Posted by Gemini View Post
    Thanks, but if my example is correct, I'll just use it. If you don't want drawings, you better tell it

    Fine. Let's hope your teacher is more understanding than I am, and let's hope your gambling on your grade pays you.

    Tonio
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  5. #5
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    I just wouldn't feel right copying your example, at least not when I could come up with something of my own.
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