That's the whole point- the equivalence relation makes 0 and 1 equivalent- you are bending [0,1] (not (0,1)) back on itself so it becomes a circle.pivoxa15 said:Homework Statement
If Z acts on R by n.x=n+x then R/Z is just S^1. CLaims the book
But I think R/Z is (0,1)
The Attempt at a Solution
Any number greater than or equal to 1 is dealt with by the equivalence relation. How does the unit circle come into it? We are dealing only with one dimensional space here.
An equivalence relation is a type of relation between two objects that follows three properties: reflexivity, symmetry, and transitivity. This means that the relation is reflexive, which means every element is related to itself, symmetric, which means if x is related to y, then y is also related to x, and transitive, which means if x is related to y and y is related to z, then x is also related to z.
In the context of the unit circle, the equivalence relation refers to the relationship between points on the circle that have the same distance from the origin. This means that any two points on the unit circle that have the same distance from the origin are considered equivalent under this relation.
R/Z, or the quotient group of the real numbers by the integers, is a mathematical concept used to describe the equivalence classes on the unit circle. Essentially, it represents the set of all possible angles on the unit circle, where each angle is related to another angle by a multiple of 2π.
R/Z and S^1 are both important concepts in mathematics because they help us understand the structure and properties of the unit circle. They also have applications in various fields such as geometry, trigonometry, and complex analysis.
Equivalence relations on the unit circle can be applied in various real-world scenarios, such as navigation, astronomy, and engineering. For example, understanding the symmetry and transitivity of points on the unit circle can help us determine the shortest distance between two points, which is useful in navigation and calculating trajectories in engineering.