The question I had to solve was: Let S be the set of real numbers. If a, b are elements of S, define a related to b if a-b is an integer. Show that the relation is an equivalence relation on S. Describe the equivalence classes of S. I was able to show that it is an equivalence relation, and so I don't need help with that. The question is what am I supposed to write for a description of the equivlance classes of S. Actually, I am able to come up with what an equivalence class looks like--it is the set of elements, say x, in the real numbers such that x=a+k for some integer k. My understanding is that the equivalence classes partition the set and that the union of all the equivalence classes would equal the entire set S. Since the equivalence classes consist of only integers, how can their union be the set of real numbers? As an example, the equivalence class of say, the square root of 3 is the empty set. Yet when you union all the equivalence classes, you will not get the real numbers, just the integers. Hope this all makes sense and that you can clear up the confusion I'm having. Thanks!
The question I had to solve was:
Let S be the set of real numbers. If a, b are elements of S, define a related to b if a-b is an integer. Show that the relation is an equivalence relation on S. Describe the equivalence classes of S.
I was able to show that it is an equivalence relation, and so I don't need help with that. The question is what am I supposed to write for a description of the equivlance classes of S. Actually, I am able to come up with what an equivalence class looks like--it is the set of elements, say x, in the real numbers such that x=a+k for some integer k. My understanding is that the equivalence classes partition the set and that the union of all the equivalence classes would equal the entire set S. Since the equivalence classes consist of only integers, how can their union be the set of real numbers? As an example, the equivalence class of say, the square root of 3 is the empty set. Yet when you union all the equivalence classes, you will not get the real numbers, just the integers. Hope this all makes sense and that you can clear up the confusion I'm having. Thanks!
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