This question came in one of class-room discussion.

Given finitely many distinct real numbers $x_1,x_2,\cdots, x_n$, does there exists a real number $y$ such that $y+x_1, y+x_2, \cdots, y+x_n$ are irrational?

(Similar question can be posed with expectation that $y+x_i$ are rational for all $i$.)

  • $\begingroup$ Yes, because $\mathbb{Q}$ is countable. No for the second question. $\endgroup$ – user296602 Feb 14 '16 at 4:42
  • $\begingroup$ Thanks. I will try to prove (I was in confusion, whether this is possible or not. Thanks for convening of truth.) $\endgroup$ – p Groups Feb 14 '16 at 4:44

HINT (if you are familiar with uncountable sets): For any $x$, how many $y$ are there such that $x+y$ is rational? How does this compare to the whole set of reals? Do you see how to bring multiple different $x$ into it?

Note, by the way, that this works even if we have infinitely many (but countably many) $x$.

You may also be interested in the Baire category theorem.

For the second question, the answer is no even for two values of $x$: do you see why $y+0$ and $y+\pi$ can never both be rational?

  • $\begingroup$ Great. Nice answer. $\endgroup$ – p Groups Feb 14 '16 at 4:45

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