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sols9 - 22 9 September 2th Quotients modulo an equivalence...

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22 9. September 2th: Quotients modulo an equivalence relation. 9.1 . (1) Define a relation 1 on R 2 by prescribing ( x 1 , y 1 ) 1 ( x 2 , y 2 ) ⇐⇒ x 1 = x 2 and y 2 - y 1 Z . (This is easily checked to be an equivalence relation.) Find a good description of the quotient R 2 / 1 . (2) Do the same for the equivalence relation 2 on R 2 defined by ( x 1 , y 1 ) 2 ( x 2 , y 2 ) ⇐⇒ x 2 - x 1 Z and y 2 - y 1 Z . Answer: This exercise was meant to let you stretch your imagination a little, after absorbing the example of the ‘real line wrapping around a circle’, Example 9.4 in the notes. In Example 9.4, the challenge was to identify together all points x R with the same decimal part, and we settled on ‘wrapping the real line around a circle’: this has the e ff ect of bringing together the points . . . 0 . 17 , 1 . 17 , 2 . 17 , 3 . 17 , . . . (for example), which is what we were trying to do. Just as in that example, in (1) the challenge is to identify together all points ( x, y ) such that y has a given decimal part, and now x is a fixed number a . So the points
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