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For a function $f$ that maps set $A$ to $B$,

  • $f\colon\mathbb R^+\to\mathbb R^+$, $f(x) = x^2$ is injective.
  • $f\colon\mathbb R\to\mathbb R$, $f(x) = x^2$ is not injective since $(- x)^2 = x^2$.

what is the difference between $\mathbb R^+$ and $\mathbb R$?

Additionally, what is the difference between $\mathbb N$ and $\mathbb N^+$?

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$\mathbb R^+$ commonly denotes the set of positive real numbers, that is: $$\mathbb R^+ = \{x\in\mathbb R\mid x>0\}$$

It is also denoted by $\mathbb R^{>0},\mathbb R_+$ and so on.

For $\mathbb N$ and $\mathbb N^+$ the difference is similar, however it may be non-existent if you define $0\notin\mathbb N$. In many set theory books $0$ is a natural number, while in analysis it is often not considered a natural number. Your mileage may vary on $\mathbb N$ vs. $\mathbb N^+$.

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  • $\begingroup$ I see now. I thought it meant that R+ included some extra element. This interpretation makes much more sense. $\endgroup$ – James Aug 26 '12 at 20:02
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    $\begingroup$ Note that $\mathbb N^+ = \mathbb Z^+$. Also if you want to confuse your readers, you can write the empty set as $\mathbb N^-$, the set of negative natural numbers. :-) $\endgroup$ – celtschk Aug 26 '12 at 20:40
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    $\begingroup$ NB : Depending on the country, $\mathbb R^+$ is also used for the set of non-negative real numbers. $\endgroup$ – Student Aug 26 '12 at 22:15
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    $\begingroup$ @Carl: Read the first line, I also say it defines the positive numbers. I just never thought zero is positive... :-) (Also, these French just have to do everything the other way around...! :-)) $\endgroup$ – Asaf Karagila Aug 27 '12 at 1:19
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    $\begingroup$ @user477343: I don't understand you, and your notation is not consistent. So I'm going to stop replying now. Have a great day! $\endgroup$ – Asaf Karagila Jan 26 '18 at 8:04
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Simply $\mathbb R$ means the set of real numbers.

$\mathbb R^+$ means the set of positive real numbers.

And $\mathbb R^-$ means the set of negative real numbers.

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