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Define an injection $(0,1)^2\rightarrow(0,1)$. Is your function surjective? Explain. Hint: use decimal expansions.

I am so confused. What does $(0,1)^2$ mean? It's not cartesian product, right? Is (0,1) a coordinate or the set of reals from 0 to 1? A hint would be much appreciated.

beginner
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    $(0,1)^2$ is the interior of the unit square, so it would consist of an ordered pair of numbers $(a,b)$, where $a\in (0,1)$ and $b\in(0,1)$. – Vasting Oct 22 '20 at 00:35
  • @Vasting is there an alternate meaning, given the context of the question? I never saw this in class before. – beginner Oct 22 '20 at 00:37
  • I am not exactly sure what you mean by this, so please let me know if you're still confused. But it is in fact the cartesian product that you mention. For example, $(0.5,0.5)\in(0,1)^2$. – Vasting Oct 22 '20 at 00:38
  • One alternate meaning is all possible products $a \cdot b$, where $a,b \in (0,1)$. Does this help? I think, that $(0,1) \times (0,1)$ is more appropriate for your question. – zkutch Oct 22 '20 at 00:38
  • This post may provide the answer you are looking for: https://math.stackexchange.com/questions/197735/injective-function-from-mathbbr2-to-mathbbr – DreiCleaner Oct 22 '20 at 00:39
  • The meaning of "$(0,1)$" in this context is the "unit open interval," i.e. the collection of all real numbers between $0$ and $1$ non-inclusive. See this: https://en.wikipedia.org/wiki/Interval_(mathematics) – gdd Oct 22 '20 at 00:42
  • @Vasting Thank you! I was confused because we didn't discuss intervals in my class yet – beginner Oct 22 '20 at 00:46
  • @GaryD Thank you, I understand – beginner Oct 22 '20 at 00:47
  • @zkutch it's not product of reals but cartesian product right. Like (.5,.5) $\in (0,1)^2$ – beginner Oct 22 '20 at 01:07
  • @beginner. You asked about alternatives, so I gave alternative variant: $(1,3) \cdot (2,4)=(2,12)$ is, depending on context, as natural as $(1,3) \times (2,4)$. You can see in my comment which interpretation, I think, is more appropriate, because in former question becomes trivial. – zkutch Oct 22 '20 at 01:43
  • @zkutch okay, ty! – beginner Oct 22 '20 at 01:47

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