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Ming Wang
Posts:
3
From:
USA
Registered:
3/25/13
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Equivalence relation
Posted:
Mar 25, 2013 3:31 PM
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Describe the resulting figure.
Let A be the set of points in the rectangle with x and y
coordinates between 0 and 1. That is,
A = {(x, y) ? R × R| 0 ? x ?1 and 0? y ? 1}.
Define a relation R on A as follows: For all (x1, y1) and (x2, y2) in A,
(x1, y1) R (x2, y2) ?
(x1, y1) = (x2, y2); or
x1 =0 and x2 =1 and y1 = y2; or
x1 =1 and x2 =0 and y1 = y2; or
y1 =0 and y2 =1 and x1 = x2; or
y1 =1 and y2 =0 and x1 = x2.
In other words, all points along the top edge of the rectangle
are related to the points along the bottom edge directly
beneath them, and all points directly opposite each other
along the left and right edges are related to each other.
The points in the interior of the rectangle are not related to
anything other than themselves. Then R is an equivalence
relation on A. Imagine gluing together all the points that
are in the same equivalence class. Describe the resulting
figure.
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