-2

For x,y∈I:=[0,1] define the relation on I as x−y∈Q.

How big (using cardinal number) is the cardinality of the equivalence class [1/√2]?

I have tried to solve it by finding the equivalence class but I'm not sure about the method that I can solve the question.

Please, can you help me to solve it.

user112363
  • 1
  • 1

1 Answers1

1

An element $x\in I$ is in the class $[1/\sqrt 2]$ iff $x - 1/\sqrt 2 \in \mathbb Q$ iff $x = 1/\sqrt 2 + q$ for $q\in \mathbb Q$. In particular, the function $x \mapsto x-1/\sqrt 2$ identifies the equivalence class $[1/\sqrt 2]$ with the set $ \mathbb Q \cap [-\frac1{\sqrt 2},1 - \frac1{\sqrt 2}] $. This is an infinite subset of $\mathbb Q$, and hence countable.

Theo Johnson-Freyd
  • 3,119