0
$\begingroup$

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 $i$.)

$\endgroup$
2
  • $\begingroup$ 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
1
$\begingroup$

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?

$\endgroup$
1
  • $\begingroup$ Great. Nice answer. $\endgroup$ – p Groups Feb 14 '16 at 4:45

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.