(5.) Consider the unit circle C with equation x + y = 1 in the plane....

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(5.) Consider the unit circle C with equation x² + y² = 1 in the plane. Thus C = {(x, y) ERX R: x² + y² = 1}. Define a relation ~ on C as follows: For (x, y) E C, (x, y) is related to itself and to its antipodal point (-x, -y). In symbols, (x, y)~ (x, y); (x, y) ~(-x, y). (a) Show that is an equivalence relation. (b) Think of C as a rubber band and imagine gluing together the equivalent antipodal points of C. Describe the resulting figure. (Much more will be said on this topic in Chapter 7.) (5.) Consider the unit circle C with equation x² + y² = 1 in the plane. Thus C = {(x, y) ERX R: x² + y² = 1}. Define a relation ~ on C as follows: For (x, y) E C, (x, y) is related to itself and to its antipodal point (-x, -y). In symbols, (x, y)~ (x, y); (x, y) ~(-x, y). (a) Show that is an equivalence relation. (b) Think of C as a rubber band and imagine gluing together the equivalent antipodal points of C. Describe the resulting figure. (Much more will be said on this topic in Chapter 7.)

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Solution consider the unit circle x y 1 in the plane c xy EIRXIR ayt1 Define a a that is every it se... View the full answer