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Ming Wang
Posts:
3
From:
USA
Registered:
3/25/13


Equivalence relation
Posted:
Mar 25, 2013 3:31 PM


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.



