540hw2 - HW2 1. Prove that any set of triangles in R 2...

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Unformatted text preview: HW2 1. Prove that any set of triangles in R 2 whose vertices have rational coor- dinates is countable. Note: Let x 1 ; x 2 ; x 3 2 R 2 . Then the triangle T with vertices x 1 ; x 2 ; x 3 is the union of points of the segments [ x 1 ; x 2 ] ; [ x 2 ; x 3 ] and [ x 1 ; x 3 ] . 2. Consider a set T of letters "T" in the plane without common points (letters "T" can be of di/erent sizes, they can be rotated and letter "T" does not have &interiorpoints, as shown in the sketch. A B C O Can such set be uncountable? Justify your answer. 3. Consider a set U of Greek letters " & " (gamma) in the plane without common points (letters " & " can be of di/erent sizes, and they can be rotated). Can such set be uncountable? Justify your answer. . . . 1 4. Prove Cantor&s theorem: let X 6 = ? . Then there is no surjective map- ping from X onto its power set 2 X . Hint: Assume such surjection f exists and consider f x 2 X j x = 2 f ( x ) g ....
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540hw2 - HW2 1. Prove that any set of triangles in R 2...

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