Equivalence Relation on R (real numbers)

Question

1. Let R be the relation on R(real numbers) defined by: For all x, y (that belong) to R(real numbers), x relates y <=> x-y (that belongs) to Z.
   (a) Is R an equivalence relation? Prove your answer.

(b) Is it true that for all a (that belongs to)R, there exists b(that belongs to)R so that b(belongs to)[a]? Explain.

(c) Is it true that for all a (that belongs to)R, there exists a rational number b so that b (belongs to)[a]? Explain.

(d) Is it true that for all a (that belongs to)R, there exists an irrational number b so that b (belongs to)[a]? Explain. For any irrational numbers other than p2, you must justify why they are irrational.

Answer 1

(a) Is it reflexive? Let $x \in \mathbb{R}$. What is $x - x$? Is $0$ an integer?

Is it symmetric? If I take $a - b$ and get an integer, does it hold that $b - a$ is also an integer? Isn't $b-a = -1 * (a - b)$? Are the integers closed under multiplication?

Is it transitive? You should be able to handle this one.

(b) Consider the reflexive case.

(c) Consider $a = \pi$.

(d) Consider $a = \frac{1}{2}$. What happens when you subtract an irrational number from $a$?