Multiple choice question about the set $A = \{ m + n\sqrt{2} \}$.

Question

let $A = \{m + n \sqrt{2}\}$ where $m,n$ are integers, then

$a.$ $A$ is dense in $R$.

$b$. $A$ has no limit point in $R$.

$c$. $A$ has only countably many limit points in $R$.

$d$. only irrational numbers can be limit point of $A$

Set $A$ is similar to the set $N \times N$, so all points are isolated in $A$.
therefore option $b$ should be correct. In answer key, option $a$ is marked as correct. Please suggest if I am doing something wrong.

You say "Set $A$ can be represented as $(m,n\sqrt2),\ldots$". — Angina Seng, Dec 27 '18 at 05:43
You now say "Set $A$ is similar to the set $\Bbb N\times\Bbb N$". Similar? How? — Angina Seng, Dec 27 '18 at 05:47

score 0 · Accepted Answer · answered Dec 27 '18 at 05:46

0

Consider $\alpha=\sqrt2-1$. Then $\alpha^n\in A$ for all $n\in\Bbb N$ (why?) and $\alpha^n\to0$. So $0$ is a limit point of $A$, which contradicts (d).

In general, if $I$ is an open interval in $\Bbb R$ then $\alpha^n$ is less than the length of $I$ for some $n\in\Bbb N$, and then $m\alpha^n\in I$ for some $m\in \Bbb Z$.

answered Dec 27 '18 at 05:46

Angina Seng

158,341

Thanks, I got it. Can't this set be similar to $Q \times Q$ or $N \times N$ in terms of cardinality? – Mathaddict Dec 27 '18 at 06:09

Multiple choice question about the set $A = \{ m + n\sqrt{2} \}$.

1 Answers1